In Part IV of the book, we turn to the questions of some of the strongly gravitating objects in the Universe and the motions that they promote.
In Chapter 20, we review the machinery of Newtonian gravity and the methods for determining the possible motions in a Newtonian gravitating potential.
In Chapter 21, we discuss the Schwarzschild metric, which describes a stationary, spherically symmetric distribution of mass. We then determine the possible motions allowed by such a metric in Chapters 22, 23 and 24.
In Chapters 25-29, we examine static black holes, objects that gravitate so strongly that not even light can escape their interior. Black holes likely contain a singularity in spacetime, which are understood using tools from Chapters 26 and 27. In Chapter 28, we discuss black hole thermodynamics and the radiation that can emerge from black holes. Finally, in Chapter 29, we discuss the rich structures that result if we allow our black holes to carry charge or to rotate in space.
^(1){ }^{1} We shall temporarily restore factors of GG in this chapter to make contact with familiar-looking equations. ^(2){ }^{2} The dot denotes a derivative with re spect to tt in this chapter. ^(3){ }^{3} The particle is at vec(r)=r vec(e)_( vec(r))\vec{r}=r \vec{e}_{\vec{r}} and note that in polar coordinates
$$
where,asadvertisedinExample 10.4,weuseorthonormalcoordinatesinthisproblem(denotedbyhatsonindices).Hence,where, as advertised in Example 10.4, we use orthonormal coordinates in this problem (denoted by hats on indices). Hence,
The moon gravitates towards the earth and by the force of gravity is continually drawn off from a rectilinear motion and retained in its orbit.
Isaac Newton
'But the Solar System!' I protested.
'What the deuce is it to me?' (Sherlock Holmes) interrupted impatiently: 'you say that we go round the sun. If we went round the moon it would not make a penny-worth of difference to me and my work.'
Arthur Conan Doyle (1859-1930) A Study in Scarlet
General relativity is well known to provide corrections to Newtonian gravitation. In this chapter, we take a step back, and present a set of methods to find and describe the trajectories allowed by the Phi(r)prop-1//r\Phi(r) \propto-1 / r potential of Newtonian gravitation. On solving the problem we shall find that there is a restricted class of possible trajectories. It turns out that the same methods employed here can be used to deal with the more varied trajectories allowed in general relativity.
In Newtonian gravitation, the potential energy due to the gravitational interaction between two particles with masses mm and MM, separated by a distance rr, is given by ^(1)U(r)=-GMm//r{ }^{1} U(r)=-G M m / r. We take the mass MM to be fixed at the origin and the mass mm to be separated from MM by the 3 -vector vec(r)\vec{r}, and moving with momentum vec(p)=m vec(v)\vec{p}=m \vec{v}. The Newtonian force on the mass mm, given by vec(F)(r)=-GMm vec(r)//r^(3)\vec{F}(r)=-G M m \vec{r} / r^{3}, acts radially, and therefore does not give rise to any torque vec(tau)\vec{\tau}. This is because vec(tau)= vec(r)xx vec(F)\vec{\tau}=\vec{r} \times \vec{F} and so, for a central force such as Newtonian gravitation, vec(tau)prop vec(r)xx vec(r)=0\vec{\tau} \propto \vec{r} \times \vec{r}=0. Since the angular momentum of the moving mass vec(L)= vec(r)xx vec(p)\vec{L}=\vec{r} \times \vec{p} is related to the torque via ^(2) vec(tau)= vec(L)^(˙){ }^{2} \vec{\tau}=\dot{\vec{L}}, the angular momentum for any central force is a constant of the motion. This constant angular momentum can be used, along with the constant energy of the system EE, to classify the possible trajectories of the moving particle.
As a further result of the conservation of angular momentum, the vector vec(L)\vec{L} (which is, by definition, perpendicular to vec(r)\vec{r} and vec(p)\vec{p} ) remains fixed. This means that the path of a particle in Newtonian gravitation can be taken as being confined to the equatorial plane of a set of threedimensional spherical coordinates. Taking vec(L)\vec{L} to be pointing along the zz-direction, we can therefore follow the paths using the two-dimensional cylindrical coordinates (r,theta)(r, \theta). In these polar coordinates, we have ^(3){ }^{3} velocity components v_( hat(r))=r^(˙)v_{\hat{r}}=\dot{r} and v_( hat(theta))=rtheta^(˙)v_{\hat{\theta}}=r \dot{\theta} and hence a squared velocity v^(2)=(r^(˙)^(2)+r^(2)theta^(˙)^(2))v^{2}=\left(\dot{r}^{2}+r^{2} \dot{\theta}^{2}\right). The system has a Lagrangian
{:(20.1)L=(1)/(2)m(r^(˙)^(2)+r^(2)theta^(˙)^(2))+(GMm)/(r).:}\begin{equation*}
L=\frac{1}{2} m\left(\dot{r}^{2}+r^{2} \dot{\theta}^{2}\right)+\frac{G M m}{r} . \tag{20.1}
\end{equation*}
Feeding this into the Euler-Lagrange equation, we obtain the two equations of motion ^(4){ }^{4}
We will meet some solutions to these equations in this chapter.
20.1 Kepler's laws
An orbit is the closed, periodic trajectory that each planet in our Solar System is, to a good approximation, observed to follow. Before Isaac Newton provided a solution to the problem of the description of orbits, Johannes Kepler ^(5){ }^{5} had formulated three laws of planetary motion resulting from his analysis of the observed trajectories of planets in our solar system. These laws are useful in understanding what the allowed orbits are. Each can be proven using the Newtonian approach, and we shall do that here. The laws are given below.
Kepler's first law: Planetary orbits follow an elliptical trajectory, with the Sun at a focus of the ellipse.
Kepler's second law: The line from Sun to planet sweeps out equal areas in equal times.
Kepler's third law: The square of the period of an orbit is proportional to the cube of the semi-major axis of the ellipse that describes its orbit.
We can immediately show how Kepler's second law is a consequence of the conservation of angular momentum.
Example 20.1
Since there is no component of the Newtonian force in the theta\theta direction, we have, from eqn 20.4, that
Integrating, we find
where CC is a constant. As shown in Fig. 20.1, CC can be interpreted geometrically as twice the rate at which the radius vector sweeps out area AA. We therefore have
This is Kepler's second law (i.e. constant A^(˙)\dot{A} ), proven from the equations of motion. As a bonus, we notice that the angular momentum is given by
This means that eqn 20.5 is a statement of vec(F)=m vec(a)\vec{F}=m \vec{a} with mm divided out. ^(5){ }^{5} Johannes Kepler (1571-1630) was plagued by ill health, complaining of myopia, multiple vision, sores, stomach and gall bladder problems, piles, rashes, mange, worms, and the delusion that he was a dog. At one point he had to defend his mother against charges of witchcraft. His work on orbits was motivated by his attempts to describe the planetary orbits in terms of plantonic solids, using data from Tycho Brahe's (1546-1601) naked-eye observations. The problem of computing the orbits of a single particle under Newtonian gravity continues to be called the Kepler problem.
Fig. 20.1 A small displacement to the radius vector vec(r)(t)\vec{r}(t) changes the vector by an amount |Delta vec(r)|=2r sin Delta theta//2~~r Delta theta|\Delta \vec{r}|=2 r \sin \Delta \theta / 2 \approx r \Delta \theta. The area of the triangle in the figure is then Delta A=(1)/(2)r^(2)Delta theta\Delta A=\frac{1}{2} r^{2} \Delta \theta, and so area is swept out at a rate A^(˙)=r^(2)theta^(˙)//2(=C//2)\dot{A}=r^{2} \dot{\theta} / 2(=C / 2).
Fig. 20.2 The ellipse is a conic section, achieved via a slice made at an angle to the base. A slice made parallel to the
Fig. 20.3 The anatomy of an ellipse, showing the foci FF and F^(')F^{\prime}, semi-major axis aa, the semi-minor axis bb and the eccentricity epsilon\epsilon. The point P\mathcal{P} is a distance rr from FF. A circular orbit is a special case where a=b,epsilon=0a=b, \epsilon=0 and FF coincides with F^(')F^{\prime}.
Taking the area to be a vector vec(A)\vec{A} parallel to vec(e)_( tilde(z))\vec{e}_{\tilde{z}}, we have
Since vec(L)\vec{L} is a constant, vec(A)^(˙)\dot{\vec{A}} is also constant. Kepler's second law therefore reflects the conservation of angular momentum, which itself follows for any central gravitational force.
20.2 Anatomy of an orbit
Kepler's first law states that orbits follow elliptical paths. The description of orbits enjoys a rich range of terminology and here we review the necessary vocabulary.
An ellipse is one of a class of two-dimensional figures known as the conic sections that also includes the circle, the parabola and the hyperbola. As the name suggests, these are figures that are generated by taking slices of cones. The slice through a cone needed to generate an ellipse is shown in Fig. 20.2 and a typical ellipse is shown in Fig. 20.3. An ellipse is described by an equation in polar coordinates
where aa and bb are, respectively, the semi-major and semi-minor axes of the ellipse and epsilon\epsilon is known as the eccentricity. [The angle theta\theta is the one that the radial vector vec(r)\vec{r} (linking the planet's position P\mathcal{P} and the focus) makes to the radial vector corresponding to the planet being at its position of closest approach to the focus, as shown in Fig. 20.3.] With an equation for the elliptical trajectory available, we can show that it is compatible with a 1//r^(2)1 / r^{2} central force.
Example 20.2
We shall investigate the nature of the acceleration of a particle following an elliptical trajectory. Differentiate the equation for an ellipse to find
{:(20.15)a_( hat(r))=(C^(2))/(r^(2))((epsilon a cos theta)/(b^(2))-(1)/(r)):}\begin{equation*}
a_{\hat{r}}=\frac{C^{2}}{r^{2}}\left(\frac{\epsilon a \cos \theta}{b^{2}}-\frac{1}{r}\right) \tag{20.15}
\end{equation*}
Using the radial acceleration a_( hat(r))=r^(¨)-rtheta^(˙)^(2)a_{\hat{r}}=\ddot{r}-r \dot{\theta}^{2} we find
Referring back to the equation for an ellipse we see that the part in the bracket is equal to ^(6)-a//b^(2){ }^{6}-a / b^{2} and so we can write
Since acceleration is proportional to force, we conclude that an elliptical orbit is compatible with a 1//r^(2)1 / r^{2} force law.
The expression for radial acceleration allows us to prove Kepler's third law.
Example 20.3
The area of an ellipse is given by A=pi abA=\pi a b. Using the notation from the previous example, the period of the orbit is T=pi ab//(C//2)T=\pi a b /(C / 2). Substituting into eqn 20.16 we have
which shows that the square of the period is proportional to the cube of aa, the semi-major axis of the ellipse.
For an object in orbit around the Sun, the perihelion is the position of closest approach to the Sun. The aphelion is the position in the orbit furthest from the Sun (Fig. 20.4). These words are often incorrectly applied to the orbits of objects around bodies other than the Sun. In fact, for (i) orbits around the Earth the points are called the perigee and apogee; (ii) for orbits around a star we call them the periastron and apastron and (iii), most generally, for orbits around any centre of mass, they are called the periapsis and apoapsis.
We can exploit the properties of the aphelion to relate the total energy of the orbiting planet to the dimensions of the ellipse.
Example 20.4
At the aphelion we have r=a(1+epsilon)r=a(1+\epsilon). Also, since at this point theta=pi\theta=\pi and so, from eqn 20.11, b^(2)=a^(2)(1-epsilon^(2))b^{2}=a^{2}\left(1-\epsilon^{2}\right). The potential energy at the aphelion for a planet in orbit around the Sun is
{:(20.19)U=-(GMm)/(a(1+epsilon)):}\begin{equation*}
U=-\frac{G M m}{a(1+\epsilon)} \tag{20.19}
\end{equation*}
At this point v_( hat(r))=0v_{\hat{r}}=0 and so the kinetic energy is
{:(20.20)K=(1)/(2)mv_( hat(theta))^(2)=(1)/(2)mr^(2)theta^(˙)^(2)=(mC^(2))/(2r^(2)):}\begin{equation*}
K=\frac{1}{2} m v_{\hat{\theta}}^{2}=\frac{1}{2} m r^{2} \dot{\theta}^{2}=\frac{m C^{2}}{2 r^{2}} \tag{20.20}
\end{equation*}
where we've used eqn 20.7 in the final step. Substituting for rr we find
{:(20.23)K=(GMm(1-epsilon))/(2a(1+epsilon)):}\begin{equation*}
K=\frac{G M m(1-\epsilon)}{2 a(1+\epsilon)} \tag{20.23}
\end{equation*}
This allows us to sum
{:[(20.24)E=K+U=(GMm(1-epsilon))/(2a(1+epsilon))-(GMm)/(a(1+epsilon))","],[(20.25)E=-(GMm)/(2a)","]:}\begin{gather*}
E=K+U=\frac{G M m(1-\epsilon)}{2 a(1+\epsilon)}-\frac{G M m}{a(1+\epsilon)}, \tag{20.24}\\
E=-\frac{G M m}{2 a}, \tag{20.25}
\end{gather*}
Fig. 20.4 The perihelion pp and aphelion aa of an orbit around the Sun S at one of the foci of the elliptical orbit. ^(7){ }^{7} Write
Since |L|=Cm|L|=C m, we can use this to deive the useful result that
L^(2)=GMm^(2)a(1-epsilon^(2)),L^{2}=G M m^{2} a\left(1-\epsilon^{2}\right),
for an elliptical orbit.
Fig. 20.5 The two contributions to the Newtonian effective potential energy (the repulsive angular momentum barrier and attractive -1//r-1 / r contribution, shown by dotted lines) give the curve shown with the solid line. ^(8){ }^{8} This term is sometimes called the angular momentum barrier. ^(9){ }^{9} It lies at the minimum because the system's energy is determined only by U_("eff ")U_{\text {eff }}.
Fig. 20.6 Energies for an elliptical orbit with energy E_(1) < 0E_{1}<0 and an unbound trajectory (E_(2) > 0)\left(E_{2}>0\right).
20.3 Effective potentials
The method of effective potentials is especially useful in understanding which trajectories are possible. We shall use it in the relativistic case and so we introduce it here. If the particle of mass mm is moving with velocity vec(v)\vec{v} in the field of the stationary mass MM then the energy of the two-particle system is written as
{:(20.26)E=(1)/(2)m(r^(˙)^(2)+r^(2)theta^(˙)^(2))-(GMm)/(r):}\begin{equation*}
E=\frac{1}{2} m\left(\dot{r}^{2}+r^{2} \dot{\theta}^{2}\right)-\frac{G M m}{r} \tag{20.26}
\end{equation*}
The angular momentum has a magnitude |L|=mtheta^(˙)r^(2)|L|=m \dot{\theta} r^{2}, which allows us to substitute for theta^(˙)\dot{\theta} and then drop the angular contribution into an effective potential energy function U_("eff ")(r)U_{\text {eff }}(r). Specifically, we write
{:(20.27)E=(1)/(2)mr^(˙)^(2)+U_(eff)(r):}\begin{equation*}
E=\frac{1}{2} m \dot{r}^{2}+U_{\mathrm{eff}}(r) \tag{20.27}
\end{equation*}
where the effective potential energy is given by
{:(20.28)U_(eff)(r)=(L^(2))/(2mr^(2))-(GMm)/(r):}\begin{equation*}
U_{\mathrm{eff}}(r)=\frac{L^{2}}{2 m r^{2}}-\frac{G M m}{r} \tag{20.28}
\end{equation*}
Given an initial angular momentum of a particle in this field, we can write the effective potential. As we shall see, it is this effective potential that can be used to classify and understand all of the trajectories that are possible for particles in the Newtonian potential. The potential energy function U_("eff ")(r)U_{\text {eff }}(r) has the two contributions shown in Fig. 20.5. There is (i) a repulsive contribution whose strength depends on the angular momentum, ^(8){ }^{8} going as 1//r^(2)1 / r^{2}; and (ii) an attractive contribution, given by the usual -1//r-1 / r potential. The difference in power law and sign means that the resultant potential can have a minimum. The limits of the effective potential for small and large rr are instructive too: (i) the effect of gravitation dies off at large distances as 1//r1 / r irrespective of the angular momentum; (ii) the potential gets very large at small rr owing to the angular momentum term. This means that any particle with non-zero angular momentum is scattered by the potential. (There is no option to spiral into the origin, for example). The only trajectory that can end up at the origin has |L|=0|L|=0, which amounts to a radial plunge into the source of the potential.
The values of rr for which E=U_(eff)(r)E=U_{\mathrm{eff}}(r) set some limits on the motion. We have this condition when r^(˙)=0\dot{r}=0. This doesn't mean that the particle has stopped (because we can still have theta^(˙)!=0\dot{\theta} \neq 0 ), rather, the particle is turning at constant rr, as it does at the points nearest and furthest from the focus of the ellipse. If r^(˙)=0\dot{r}=0 at all times, then the trajectory must be circular (because there is never a change in rr ) with a radius r_(0)r_{0} that corresponds to the minimum ^(9){ }^{9} of U_("eff ")U_{\text {eff }}. This occurs when delU_("eff ")(r)// del r|_(r=r_(0))=0\partial U_{\text {eff }}(r) /\left.\partial r\right|_{r=r_{0}}=0 or
{:(20.29){:(L^(2))/(mr_(0))=GMm quad" (circular orbit "):}\begin{equation*}
\left.\frac{L^{2}}{m r_{0}}=G M m \quad \text { (circular orbit }\right) \tag{20.29}
\end{equation*}
The circular orbit has a negative total energy E=-GMm//2r_(0)E=-G M m / 2 r_{0}, so that the orbiting particle is in a bound state that requires energy to
be inputted in order to escape. There are other possible bound orbits with E < 0E<0, where the particle can be thought of as being confined by the effective potential, bouncing back and forth between the two values where E=U_("eff ")^(10)E=U_{\text {eff }}{ }^{10} These are, of course, the ellipses that Kepler's first law describes. We can immediately gain some insight by looking at the negative values of U_("eff ")(r)U_{\text {eff }}(r), since E=U_("eff ")E=U_{\text {eff }} for the elliptical paths at the perihelion and aphelion, where r^(˙)=0\dot{r}=0. As shown in Fig. 20.6, lines of negative constant total energy EE intersect U_("eff ")(r)U_{\text {eff }}(r) at two points when r^(˙)=0\dot{r}=0, providing the length of semi-major and semi-minor axes.
In contrast, an unbounded trajectory (i.e. not an orbit) with a positive energy has a unique distance of closest approach to the gravitating mass. At this position r^(˙)=0\dot{r}=0 and so the distance of closest approach occurs when the energy of the system E=U_("eff ")(r)E=U_{\text {eff }}(r). We can therefore use the graph of the effective potential energy (Fig. 20.6) to describe all of the trajectories. We imagine the particle moving horizontally as its value of rr varies; it is deflected whenever it meets the curve U_("eff ")U_{\text {eff }}. Remember that the form of U_("eff ")U_{\text {eff }} is determined by the (constant) angular momentum of the particle. We shall use this graphical method again when we look into the trajectories allowed by general relativity.
20.4 Allowed trajectories
We can now solve the problem once and for all, by determining all of the possible motions in the Newtonian potential. We have seen that orbits are expected in a Newtonian potential and that these have a negative total energy EE. If the energy is positive, then we expect the trajectory to be unbounded. We shall now solve the equations of motion for the trajectories of the particles and then use the sign of the total energy to classify the orbits. The technique of choice employs the variable u=1//ru=1 / r, which we shall also use in solving the relativistic problem.
Example 20.5
What is the equation of motion for the trajectories? It's convenient to use u=1//ru=1 / r and write
{:[F_( hat(r))=-GMmu^(2)=m[(d^(2)r)/((d)t^(2))-r(((d)theta)/((d)t))^(2)]],[(20.32)=-(L^(2))/(m)u^(2)((d^(2)u)/((d)theta^(2))+u)]:}\begin{align*}
F_{\hat{r}}=-G M m u^{2} & =m\left[\frac{\mathrm{~d}^{2} r}{\mathrm{~d} t^{2}}-r\left(\frac{\mathrm{~d} \theta}{\mathrm{~d} t}\right)^{2}\right] \\
& =-\frac{L^{2}}{m} u^{2}\left(\frac{\mathrm{~d}^{2} u}{\mathrm{~d} \theta^{2}}+u\right) \tag{20.32}
\end{align*}
We end up with a differential equation for the trajectories
{:(20.33)(d^(2)u)/((d)theta^(2))+u=(GMm^(2))/(L^(2)):}\begin{equation*}
\frac{\mathrm{d}^{2} u}{\mathrm{~d} \theta^{2}}+u=\frac{G M m^{2}}{L^{2}} \tag{20.33}
\end{equation*}
The solution to this equation is a function u(theta)u(\theta) that gives the trajectory. ^(10){ }^{10} The fact that the potential has a minimum implies the motion for particles bound in the potential is stable. ^(11){ }^{11} The parabola corresponds to the case epsilon=1\epsilon=1 in eqn 20.11.
Rewriting the differential equation from the last example, we use the variable u_(0)=(GMm^(2))/(L^(2))u_{0}=\frac{G M m^{2}}{L^{2}}, and end up with an equation of motion for the variable uu which is
The general solution of this equation, that gives access to all of the allowed trajectories, is
{:(20.35)u-u_(0)=B cos theta:}\begin{equation*}
u-u_{0}=B \cos \theta \tag{20.35}
\end{equation*}
where BB is a constant.
Now for some interpretation. If we can write the expression for the trajectories in terms of the total energy EE, then we can classify orbits. If the energy is negative we have a bound state or, in other words, an orbit. If the energy is positive, the trajectory is not bounded.
We shall rewrite eqn 20.35 in terms of the values of several physical quantities evaluated at the perihelion (i.e. when theta=0\theta=0 ). At this point, the radius is r=r_(1)r=r_{1} (and so u=u_(1)u=u_{1} ) the kinetic energy and potential energy are K_(1)K_{1} and U_(1)U_{1} respectively and angular momentum is LL. At the perihelion, the radius is perpendicular to the velocity and so the angular momentum is L=mv_( hat(theta))r_(1)L=m v_{\hat{\theta}} r_{1}, and so K_(1)=L^(2)//2mK_{1}=L^{2} / 2 m. This allows us to say
{:(20.36)u_(0)=(GMm^(2))/(L^(2))=-(U_(1))/(2K_(1))*u_(1).:}\begin{equation*}
u_{0}=\frac{G M m^{2}}{L^{2}}=-\frac{U_{1}}{2 K_{1}} \cdot u_{1} . \tag{20.36}
\end{equation*}
Since when u=u_(1)u=u_{1} we have theta=0\theta=0, eqn 20.35 gives us an expression u_(1)-u_(0)=Bu_{1}-u_{0}=B, which, together with the total energy E=K_(1)+U_(1)E=K_{1}+U_{1}, allows us to write and equation linking the constant total energy to the parameters u_(0)u_{0} and BB
Using this expression, we can classify the three general types of trajectory that are possible, via the sign of EE.
When B=u_(0)B=u_{0}, we have E=0E=0 and the trajectory is the parabola u=u_(0)(1+cos theta)u=u_{0}(1+\cos \theta). This allows rr to go to infinity when theta=pi\theta=\pi so the trajectory is, only just, unbounded. ^(11){ }^{11}
When B > u_(0)B>u_{0}, we have EE positive and the trajectory is a hyperbola. This allows rr to go to infinity when theta=cos^(-1)(-u_(0)//B)\theta=\cos ^{-1}\left(-u_{0} / B\right). This is an unbounded trajectory.
Finally, in order to have EE negative (resulting in a bounded orbit), we must have B < u_(0)B<u_{0}. Comparing u-u_(0)=B cos thetau-u_{0}=B \cos \theta to eqn 20.11 for the ellipse, we see that it is identical if we take u_(0)=a//b^(2)u_{0}=a / b^{2} and the eccentricity of the ellipse is given by B//u_(0)=epsilonB / u_{0}=\epsilon. The equation u-u_(0)=B cos thetau-u_{0}=B \cos \theta with B < u_(0)B<u_{0} then describes an ellipse with limiting radii r=(u_(0)+B)^(-1)r=\left(u_{0}+B\right)^{-1} and r=(u_(0)-B)^(-1)r=\left(u_{0}-B\right)^{-1}. (The case of the circular orbit corresponds to B=0B=0, as it must.)
Example 20.6
Another useful route to the finding the possible trajectories is to start with the equation for the total energy
{:(20.38)E=(1)/(2)mr^(˙)^(2)+(1)/(2)mr^(2)theta^(˙)^(2)-(GMm)/(r):}\begin{equation*}
E=\frac{1}{2} m \dot{r}^{2}+\frac{1}{2} m r^{2} \dot{\theta}^{2}-\frac{G M m}{r} \tag{20.38}
\end{equation*}
and divide through by L^(2)//2mL^{2} / 2 m. Writing u^(')=del u//del thetau^{\prime}=\partial u / \partial \theta we obtain, after a little algebra, that
{:(20.39)(u^('))^(2)+u^(2)-(2GMm^(2))/(L^(2))*u-(2Em)/(L^(2))=0:}\begin{equation*}
\left(u^{\prime}\right)^{2}+u^{2}-\frac{2 G M m^{2}}{L^{2}} \cdot u-\frac{2 E m}{L^{2}}=0 \tag{20.39}
\end{equation*}
Spotting the presence of u_(0)=1//r_(0)=GMm^(2)//L^(2)u_{0}=1 / r_{0}=G M m^{2} / L^{2} we choose to rewrite this quadratic equation as
where u_(0)^(2)(1-epsilon^(2))=-2Em//L^(2)u_{0}^{2}\left(1-\epsilon^{2}\right)=-2 E m / L^{2}. Try a solution u=c+d cos thetau=c+d \cos \theta, and find c=u_(0)c=u_{0} and d=+-u_(0)epsilond= \pm u_{0} \epsilon, so that we have
{:(20.41)u=u_(0)(1+-epsilon cos theta):}\begin{equation*}
u=u_{0}(1 \pm \epsilon \cos \theta) \tag{20.41}
\end{equation*}
With our definition of theta\theta the equation for the ellipse takes the positive sign and we have u=u_(0)(1+epsilon cos theta)u=u_{0}(1+\epsilon \cos \theta).
20.5 The why? of orbits
What guarantees that there are orbits at all? That is, why should any trajectory close and hence be periodic? Let's temporarily examine nonrelativistic orbits more generally, taking the central potential energy to be U(r)U(r), which is not necessarily the Newtonian potential U(r)prop-1//rU(r) \propto-1 / r. (In any case, the angular momentum vec(L)\vec{L} is still conserved owing to the lack of torque from a central field.) We can rewrite eqn 20.27 for the total energy as
or, since dtheta=Ldt//mr^(2)\mathrm{d} \theta=L \mathrm{~d} t / m r^{2}, we have
{:(20.44)theta=int_(r_(1))^(r_(2))((d)rL//r^(2))/({2m[E-U(r)]-(L^(2))/(r^(2))}^((1)/(2))):}\begin{equation*}
\theta=\int_{r_{1}}^{r_{2}} \frac{\mathrm{~d} r L / r^{2}}{\left\{2 m[E-U(r)]-\frac{L^{2}}{r^{2}}\right\}^{\frac{1}{2}}} \tag{20.44}
\end{equation*}
If the motion has two limiting radii, r_(min)r_{\min } and r_(max)r_{\max } then, during the time in which rr varies from r_(min)r_{\min } to r_(max)r_{\max } and back, the radius vector turns through an angle
{:(20.45)Delta theta=2int_(r_(min))^(r_(max))(drL//r^(2))/({2m[E-U(r)]-(L^(2))/(r^(2))}^((1)/(2))).:}\begin{equation*}
\Delta \theta=2 \int_{r_{\min }}^{r_{\max }} \frac{\mathrm{d} r L / r^{2}}{\left\{2 m[E-U(r)]-\frac{L^{2}}{r^{2}}\right\}^{\frac{1}{2}}} . \tag{20.45}
\end{equation*}
^(12){ }^{12} The amount of precession per orbit is given by delta theta=Delta theta-2pi\delta \theta=\Delta \theta-2 \pi.
Fig. 20.7 The precession of an orbit After nn periods we must make mm complete revolutions in order for the orbit to close. ^(13){ }^{13} Alternatively, the integral is carried out on page 30 of the book by Zee.
Fig. 20.8 The effective potential for three cases: (a) n < -3n<-3, (b) -3 < n <-3<n< -1 , (c) n > -3n>-3. Stable circular orbits at r=r_(0)r=r_{0} are only possible if n > -3n>-3.
For this motion to form a closed orbit we must have Delta theta=2pi q//n\Delta \theta=2 \pi q / n with qq and nn integers. That is to say, after nn periods of motion between r_("min ")r_{\text {min }} and r_("max ")r_{\text {max }}, the radius vector has made qq complete revolutions in theta\theta and we're back where we started. This allows for the precession of the orbits, as shown in Fig. 20.7. ^(12){ }^{12}
Example 20.7
We can demonstrate that the orbit closes for the Newtonian case, that is, when U(r)=-(alpha )/(r)U(r)=-\frac{\alpha}{r}. Rewrite the integral as
{:(20.47)Delta theta=-2(del)/(del L)int_(r_(min))^(r_(max))dr{2mE+(2m alpha)/(r)-(L^(2))/(r^(2))}^((1)/(2)):}\begin{equation*}
\Delta \theta=-2 \frac{\partial}{\partial L} \int_{r_{\min }}^{r_{\max }} \mathrm{d} r\left\{2 m E+\frac{2 m \alpha}{r}-\frac{L^{2}}{r^{2}}\right\}^{\frac{1}{2}} \tag{20.47}
\end{equation*}
This can be rewritten as an elliptical integral whose result can be looked up: ^(13){ }^{13} it turns out to yield Delta theta(=2pi q//n)=2pi\Delta \theta(=2 \pi q / n)=2 \pi. This implies that q=nq=n and there is no precession of any Newtonian orbit.
Another way of looking at the closing of trajectories is covered in the following example.
Example 20.8
For a central field of force F=-Ar^(n)F=-A r^{n}, the effective potential is
{:(20.48)V_(eff)=(Ar^(n+1))/(n+1)+(L^(2))/(2mr^(2)):}\begin{equation*}
V_{\mathrm{eff}}=\frac{A r^{n+1}}{n+1}+\frac{L^{2}}{2 m r^{2}} \tag{20.48}
\end{equation*}
This gives a stable circular orbit at r=r_(0)r=r_{0} only if n > -3n>-3. (The form of the effective potential is shown in Fig. 20.8 for three distinct cases.) Near a circular orbit at r=r_(0)r=r_{0}, we can write the effective potential as a Taylor expansion
where d^(2)V_("eff ")//dr^(2)\mathrm{d}^{2} V_{\text {eff }} / \mathrm{d} r^{2} evaluated at r=r_(0)r=r_{0} is given by (n+3)L^(2)//(mr_(0)^(4))(n+3) L^{2} /\left(m r_{0}^{4}\right). Small oscillations for a particle of mass mm near the bottom of the well, i.e. orbiting close to r=r_(0)r=r_{0}, are therefore governed by
{:(20.50)mr^(¨)+((n+3)L^(2))/(mr_(0)^(4))*(r-r_(0))=0:}\begin{equation*}
m \ddot{r}+\frac{(n+3) L^{2}}{m r_{0}^{4}} \cdot\left(r-r_{0}\right)=0 \tag{20.50}
\end{equation*}
which is simple harmonic motion with period tau\tau given by
For the n=-2n=-2 (Newtonian) case, tau=T\tau=T and so the precession leads to elliptical orbits. Another stable case occurs for n=1n=1 (simple harmonic motion) where tau=T//2\tau=T / 2.
For general n > -3n>-3, excepting these special cases, the precession tau\tau is incommensurate with the orbital motion. For this reason, any departures from n=-2n=-2 in some imagined small departure from Newtonian gravity should be easy to determine from observations since it will lead to precession of elliptical orbits.
The closing of trajectories is further demonstrated for Newtonian orbits in the next example.
Example 20.9
We start by proving a geometric result. The Laplace-Runge-Lenz vector ^(14){ }^{14} is defined as
Now consider the orbit at the perihelion (and aphelion). Here we have that the momentum vec(p)\vec{p} is perpendicular to the semi-major axis of the ellipse. Since vec(L)\vec{L} points out of the plane of the orbit we can take vec(L)\overrightarrow{\mathcal{L}} to point from the focus to the perihelion. Since the vector vec(L)\overrightarrow{\mathcal{L}} is constant, it always points to the perihelion and so the perihelion cannot move. The orbit never precesses and must therefore close. ^(15){ }^{15}
With this review of Newtonian orbits under our belts, we can examine some of the richness afforded by general relativity's correction to the Newtonian picture. In the next chapter, we meet a metric field analogous to the spherically symmetric 1//r1 / r potential of the Newtonian case. This is the celebrated Schwarzschild metric.
Chapter summary
Orbits in a Newtonian potential are elliptical. They can be understood by identifying an effective potential.
The variable u=1//ru=1 / r allows us to rewrite the problem and solve the equations of motion. We make use of conserved energy and angular momentum.
Newtonian orbits never precess. ^(14){ }^{14} Pierre-Simon Laplace (1749-1827), Carl Runge (1856-1927) and Wilhelm Lenz (1888-1957). The vector is also known as the Laplace vector, the Runge-Lenz vector and the Lenz vector. In fact, the quantity seems to predate all of these people. Herbert Goldstein's two short articles [Am. J. Phys, 43, 737 (1975) and 44, 1123 (1976)] 43, 737 (1975) and 44, 1123 (1976)] outline its interesting history, where it is suggested that priority actually beongs to Jakob Hermann (1678-1733) and Johann Bernoulli (1667-1748). As discussed in the book by Gutzwiller Wolfgang Pauli used the properties of vec(L)\overrightarrow{\mathcal{L}} to compute the quantum-mechanical spectrum of the hydrogen atom exactly in 1926. ^(15){ }^{15} The components of the vector vec(L)\overrightarrow{\mathcal{L}}, along with the energy EE, and the components of the angular momentum vec(L)\vec{L} give us seven quantities. In the Kepler give us seven quantities. vec(vec())\overrightarrow{\vec{~}} the Kepler problem, the length of L\mathcal{L} is constant and we also have L* tilde(L)=0\mathcal{L} \cdot \tilde{L}=0, giving us five independent constants of the motion. In general, a mechanical systems with dd degrees of freedom can have, at most, 2d-12 d-1 constants of the motion. [This is because there are 2d2 d initial conditions (the components of position and velocity) and the initial time cannot be determined by a constant of the motion.] The Kepler problem has d=3d=3, so we have the maximum number of constants of the motion possible. For this reason, the Kepler problem is sometimes called 'maximally superintegrable'.
Exercises
(20.1) Fill in the algebra leading to eqn 20.39 .
(20.2) Two particles, each of mass mm, move under the influence of their mutual gravitational attraction -(Gm^(2))/(r^(2))-\frac{G m^{2}}{r^{2}}. Initially, the particles are a large distance apart and approach each other with velocities vec(v)\vec{v} and - vec(v)-\vec{v} along parallel paths a distance bb apart.
(a) What is the angular momentum of the system during the motion?
(b) Write down the energy of the system at the start of the motion and at the point in the motion when the particles are closest to each other.
(c) Show that the least distance dd between the particles in their subsequent motion is given by
(20.3) A particle with mass mm is placed a distance xx from the centre of a thin ring of radius aa, along the line through the centre of the ring and perpendicular to its plane.
(a) Assuming the ring has a total mass MM, which is uniformly distributed along the ring, show that the total gravitational potential energy of the ring and particle is given by
{:(20.54)U=-(GMm)/((a^(2)+x^(2))^((1)/(2))):}\begin{equation*}
U=-\frac{G M m}{\left(a^{2}+x^{2}\right)^{\frac{1}{2}}} \tag{20.54}
\end{equation*}
(b) Find the magnitude and direction of the force on the particle. Comment on this result in the case that the distance between the particle and ring is very large compared to the ring's radius.
Now consider a mass Omega\Omega uniformly distributed over a disc of radius LL. A particle of mass mm is placed a distance xx from the centre of the disc, along the line through its centre and perpendicular to its plane.
(c) Using the result from part (a), or otherwise,
find the gravitational potential energy of the disc and particle system.
(20.4) (a) Show that an equation of motion
where Phi\Phi is the time-independent Newtonian potential, is incompatible with the rule from special relativity that a*u=0\boldsymbol{a} \cdot \boldsymbol{u}=0 (where a\boldsymbol{a} is acceleration and u\boldsymbol{u} is velocity).
(b) Show that the equation of motion
does not suffer from the problem in (a).
The latter can be regarded as an alternative theory of gravity in which the gravitational field exists in Minkowski spacetime. The projection operator (eta+u ox u)(\boldsymbol{\eta}+\boldsymbol{u} \otimes \boldsymbol{u}) picks out the part perpendicular to u\boldsymbol{u} (i.e. perpendicular to the world line, ensuring the orthogonality of this part and u\boldsymbol{u} ).
(20.5) Consider the equation of motion from Exercise 20.4(b). By taking the limit of zero mass, such that lambda=tau//m\lambda=\tau / m remains constant as mm and tau\tau go to zero, show that p^(alpha)e^(Phi(r))p^{\alpha} \mathrm{e}^{\Phi(r)} (for alpha=0-4\alpha=0-4 ) remains constant along the world line of a photon. This approach (based on that of Thorne and Blandford) shows that light cannot be deflected by the Sun in this theory, in contradiction to experiment. Gravity cannot, therefore, simply be incorporated into special relativity as another field in Minkowski spacetime.
The Schwarzschild geometry
As you see the war treated me kindly enough, in spite of heavy gunfire, to allow me to get away from it all and take this walk in the land of your ideas.
Karl Schwarzschild (1873-1916) in a 1915 letter to Einstein
After Einstein wrote down the field equation of general relativity he did not expect it to admit exact solutions owing to its complexity. He himself used an approximate solution in his 1915 article about the perihelion of Mercury. It therefore came as something of a surprise when Einstein received a letter from Karl Schwarzschild at the end of 1915 detailing a rather simple exact solution. ^(1){ }^{1} Schwarzschild was, despite being over forty years old, then serving as a soldier in World War I and carried out his work while at the Russian front.
The Schwarzschild solution is a solution of the Einstein field equation for the geometry outside a spherically symmetric, gravitating mass distribution. (We will find that the solution inside a mass distribution is different.) The Schwarzschild solution gives us a useful metric tensor that we will use to describe the spacetime outside stars and black holes, along with the motion of objects orbiting these bodies. First, the answer: the Schwarzschild metric line element for the space outside a static, spherically symmetric gravitating body of mass MM at the origin of a set of coordinates (t,r,theta,phi)(t, r, \theta, \phi) is written as ^(2){ }^{2}
Notable features of this line element include: (i) it is static: none of the components ^(3)g_(mu nu){ }^{3} g_{\mu \nu} of the metric field depend on time; (ii) it is asymptotically flat: as r rarr oor \rightarrow \infty it looks like the Minkowski metric; and (iii) it appears badly behaved at the origin, and also when r=2Mr=2 M, where the second term becomes singular. One additional reason why this line element is so special is that Birkhoff's theorem ^(4){ }^{4} tells us that any spherically symmetric solution to the Einstein equation outside a gravitating object (they don't need to be static, for example) will be identical to Schwarzschild's static solution.
In this chapter, we will examine where the Schwarzschild solution comes from, and how we justify its form. Our results will be useful in the later chapters in this part of the book, where we will apply the
21.1 Justifying the solution 230 21.2 Components of the Riemann tensor 231 21.3 A gravitating object 232 21.4 The meaning of the coordinates
Chapter summary 235
Exercises 235 ^(1){ }^{1} In 1915, Schwarzschild started suffering from pemphigus, a rare autoimmune skin disease that likely led to his death in 1916. English speakers should consider saying 'shvarts-shilt' rather consider saying shvarts-shilt rather
than the commonly heard 'shwortschild'. The name means black shield rather than black child, in any case. ^(2){ }^{2} Our choice of units in this part of the book is G=c=1G=c=1. To obtain real-world units in the metric substitute M rarr GM//c^(2)M \rightarrow G M / c^{2} and t rarr ctt \rightarrow c t. ^(3){ }^{3} The non-zero components of the metric tensor are
^(4){ }^{4} George David Birkhoff (1884-1944). Birkhoff's theorem says that any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat. ^(5){ }^{5} We'll postpone examining the light cone structure of this metric field until then.
In non-natural units, the Schwarzschild metric is given by
Note that, unlike the coordinate systems we met in the previous part of the book, Schwarzschild coordinates are book, Schore since !=0\neq 0 ar not comoving, since g_(tt,r)!=0g_{t t, r} \neq 0. Physi cally, this is because the Schwarzschild coordinates privilege the rest frame of the spherically symmetric mass distribution that acts as the source of the curvature. ^(6){ }^{6} Recall our slogan that coordinate have no intrinsic significance in the metric field, so one radius-like variable is just as good as another. ^(7){ }^{7} This also provides us with an opera tional definition of when a field can be characterized as being weak (i.e. restorcharacterized as being weak (i.e. re
ing factors, we want 2Phi//c^(2)≪12 \Phi / c^{2} \ll 1 ).
metric and use the curvature tensor to examine motion such as orbits and objects like black holes. ^(5){ }^{5}
21.1 Justifying the solution
We seek a solution to the Einstein equation for a static, spherically symmetric mass distribution centred on the origin of a set of spherical coordinates (t,r,theta,phi)(t, r, \theta, \phi). Although, as we have discussed, the names of these components have no intrinsic metric significance, it will be helpful to give them some temporarily in an effort to make sensible assumptions. We start by assuming that the metric field g\boldsymbol{g} is static. This means that intervals between events are time independent and so components obey dg_(alpha beta)//dt=0\mathrm{d} g_{\alpha \beta} / \mathrm{d} t=0. We assume that the spherical symmetry means that world lines of constant r,thetar, \theta and phi\phi are orthogonal to t=t= (const.) hypersurfaces. We also assume asymptotic flatness, which is to say that as r rarr oor \rightarrow \infty, we must have flat spacetime (that is, the gravitational forces vanish at infinity).
A good place to start is therefore with spherically symmetric, flat spacetime, which obeys the above assumptions and has a metric line element
where dOmega^(2)=dtheta^(2)+sin^(2)thetadphi^(2)\mathrm{d} \Omega^{2}=\mathrm{d} \theta^{2}+\sin ^{2} \theta \mathrm{~d} \phi^{2}. The simplest way to proceed is then to guess that these components assume different values close to the gravitating object, subject to obeying the constraints of the previous paragraph. Owing to spherical symmetry, the new components can only depend on the radius coordinate rr and so we write an expression with three functions
We can immediately now restrict the number of variables from three to two. This is because we can simply reinterpret R(r)R(r) as the rr coordinate and rescale the other functions. ^(6){ }^{6} So we have a general line element
where we constrain Phi(oo)=Lambda(oo)=0\Phi(\infty)=\Lambda(\infty)=0.
Expressions for Phi(r)\Phi(r) and Lambda(r)\Lambda(r) will come from linking the line element in eqn 21.5 to the physics of gravitating objects, which we do below. However, we can immediately gain an insight into the function Phi(r)\Phi(r) by comparison with the weak-field metric, which has a line element, expressed in spherical coordinates, of
where Phi\Phi is the Newtonian gravitational potential. In eqn 21.5, we observe that in the limit of small Phi\Phi we have that -e^(2Phi)~~-(1+2Phi)-\mathrm{e}^{2 \Phi} \approx-(1+2 \Phi), suggesting that this is the same gravitational potential Phi\Phi that features in the weak-field metric. ^(7){ }^{7}
Now that we have, in eqn 21.5, a candidate metric, we can feed it through the Einstein field equation G=8pi T\boldsymbol{G}=8 \pi \boldsymbol{T}. This allows us to discover that the metric does indeed solve the problem and also how to provide expressions for Phi\Phi and Lambda\Lambda in terms of physically meaningful quantities.
21.2 Components of the Riemann tensor
We start by finding the left-hand side of the Einstein field equation. This involves finding the curvature tensor R\boldsymbol{R} for the metric field in eqn 21.5 and then generating the Einstein tensor G\boldsymbol{G}. The components of R\boldsymbol{R} can be generated in a number of ways. To maximize simplicity, we will work in the orthonormal frame. ^(8){ }^{8}
Example 21.1
Written in its general form, the spherically symmetric metric line element looks like
where Phi\Phi and Lambda\Lambda are functions of rr only. Feeding this into the equations to find the components of the Riemann tensor, we obtain ^(9){ }^{9}
and we have used the dash notation for derivatives with respect to rr.
These components of the Riemann tensor R\boldsymbol{R} give us the non-zero components of the Einstein tensor ^(10){ }^{10}
Note that G\boldsymbol{G} will vanish outside a mass distribution, although we can have curvature (i.e. non-zero R\boldsymbol{R} ) this region. The tensor G\boldsymbol{G} is zero here because the Einstein equation forces G\boldsymbol{G} to be proportional to the energy-momentum tensor T\boldsymbol{T}, which itself vanishes in the vacuum. ^(8){ }^{8} Remember that one advantage of this is that components can be raised and lowered with the Minkowski tensor eta\boldsymbol{\eta}. The vielbein components for observers in the stationary orthonormal frame are
In the Schwarzschild case, we can safely call the observer in the this frame stationary, since we can say they are stationary relative to the mass at the origin. ^(9){ }^{9} See the exercises at the end of Chapter 11 and also Chapter 36 . ^(10){ }^{10} The simplest way to compute these is to use the useful rules that G_(0)^(0)=-(R^(12)_(12)+R^(23)_(23)+R^(31)_(31))G_{0}^{0}=-\left(R^{12}{ }_{12}+R^{23}{ }_{23}+R^{31}{ }_{31}\right), G^(1)_(1)=-(R^(02)_(02)+R^(03)_(03)+R^(23)_(23))G^{1}{ }_{1}=-\left(R^{02}{ }_{02}+R^{03}{ }_{03}+R^{23}{ }_{23}\right), G^(0)_(1)=R^(02)_(12)+R^(03)_(13)G^{0}{ }_{1}=R^{02}{ }_{12}+R^{03}{ }_{13}, G^(1)_(2)=R^(10)_(20)+R^(13)_(23)G^{1}{ }_{2}=R^{10}{ }_{20}+R^{13}{ }_{23},
where other components can be found using cyclic permutations. You're invited to prove these rules in Exercise 21.3.
This gives us the left-hand (geometrical) side of the Einstein equation. In the next section, we look at the right-hand (physical) side. ^(11){ }^{11} That is, in the orthonormal frame with basis vectors e_( hat(t)),e_( hat(r)),e_( hat(theta)),e_( hat(phi))\boldsymbol{e}_{\hat{t}}, \boldsymbol{e}_{\hat{r}}, \boldsymbol{e}_{\hat{\theta}}, \boldsymbol{e}_{\hat{\phi}} we have
Now for the right-hand side of the Einstein equation. We make a spherically symmetric distribution of static perfect fluid so that, in the orthonormal frame, the inside of the mass distribution is described by T^( hat(mu) hat(nu))=diag(rho,p,p,p)T^{\hat{\mu} \hat{\nu}}=\operatorname{diag}(\rho, p, p, p), where rho\rho is the mass density and pp is the pressure. ^(11){ }^{11} We'll take the total mass of the distribution giving rise to the metric to be MM, and stipulate that the mass distribution stretches out to some maximum radius r=Rr=R. Outside this radius there is no matter and so the components of T\boldsymbol{T} must vanish locally.
The function Lambda(r)\Lambda(r) can be determined by considering the 00th component of the Einstein equation. We have, from the previous section, that the component G_( hat(t) hat(t))=8piT_( hat(t) hat(t))G_{\hat{t} \hat{t}}=8 \pi T_{\hat{t} \hat{t}} can be written as
{:(21.16)m(r)=int_(0)^(r)dr4pir^(2)rho+m(0):}\begin{equation*}
m(r)=\int_{0}^{r} \mathrm{~d} r 4 \pi r^{2} \rho+m(0) \tag{21.16}
\end{equation*}
Since rho\rho is a density, this expression makes physical sense for a spherically symmetrical object if we interpret m(r)m(r) to be the mass contained in a sphere of radius rr.
In order to find Phi(r)\Phi(r), we need only consider the hat(r) hat(r)\hat{r} \hat{r} component of the Einstein equation, which gives us
This looks complicated, but if we concentrate on the field outside of the mass distribution, were p=0p=0, then we can solve eqn 21.19 straightforwardly. In the region r > Rr>R, we deduce from eqn 21.16 that the mass function m(r > R)m(r>R) must be constant. This constant must, of course, be the total mass MM and so we exchange m(r)m(r) for MM, set p=0p=0 and write an expression valid for outside the mass distribution of
The solution to the latter expression is ^(14){ }^{14}
{:(21.22)Phi(r)=(1)/(2)ln(1-(2M)/(r))quad" (outside the mass). ":}\begin{equation*}
\Phi(r)=\frac{1}{2} \ln \left(1-\frac{2 M}{r}\right) \quad \text { (outside the mass). } \tag{21.22}
\end{equation*}
The resulting expressions for e^(2Lambda)\mathrm{e}^{2 \Lambda} and e^(2Phi)\mathrm{e}^{2 \Phi} give us all we need to justify the form of the Schwarzschild line element outside a gravitating body, which is ^(15){ }^{15}
We also observe that components of G\boldsymbol{G} vanish as they must, since we're in a region with rho=p=0\rho=p=0 and so 0=8pi T=G0=8 \pi \boldsymbol{T}=\boldsymbol{G}.
Example 21.3
The arguments in this section can also be used to derive the Tolman-Oppenheimer-Volkov (TOV) equations that describe the structure of a static, spherical perfect fluid, and which therefore provides us with a relativistic model of a star. Starting with the stress-energy tensor for a perfect fluid, we can use grad*T=0\boldsymbol{\nabla} \cdot \boldsymbol{T}=0 to show ^(17){ }^{17} that in the Schwarzschild geometry
The second TOV equation gives us a quantity to be interpreted as the mass
{:(21.29)m(r)=4piint_(0)^(r)drr^(2)rho(r):}\begin{equation*}
m(r)=4 \pi \int_{0}^{r} \mathrm{~d} r r^{2} \rho(r) \tag{21.29}
\end{equation*}
where we have fixed the integration constant m(0)=0m(0)=0. The solution to these equations requires a choice of equation of state (that is, a link between pp and rho\rho ). They are generally integrated numerically from the origin outwards until p(r=R)=0p(r=R)=0, which we take to give the surface of the star, with m(R)m(R) giving us the stellar mass. ^(14){ }^{14} This gives us
where a dash denotes a derivative with respect to the rr coordinate.
(These are computed in Exercise 9.7). ^(15){ }^{15} In a letter to Schwarzschild in 1916, Einstein wrote "I had not expected that one could formulate the exact solution of the problem so simply. The analytcal treatment of the problem seems to me to be excellent." ^(16){ }^{16} Achieved by lowering indices using the Minkowski tensor (i.e. lowering a hat(t)\hat{t} gives a minus sign, lowering everything else has no effect).
Richard C. Tolman (1881-1948) J. Robert Oppenheimer (1904-1967). George Volkoff (1914-2000).
Oppenheimer and Volkoff used Tolman's work as a basis that led to their prediction of the existence of neutron stars. ^(17){ }^{17} See Exercise 39.5. ^(18){ }^{18} As examined in the exercises, the Newtonian prediction for hydrostatic equilibrium is
It will be useful in later chapters to have in mind a simple picture of stellar evolution. Stars are formed from gas clouds that collapse under gravity to eventually achieve equilibrium where gravitational collapse is balanced against outward radiation pressure resulting from nuclear fusion of hydrogen in the star's core. The hydrogen in the core eventually becomes exhausted, causing many stars (i.e. those of around one solar mass M_(o.)M_{\odot} ), to fuse helium inside their core and hydrogen outside, leading to expansion into a red giant leading the nuclear fuel is a red giant Once the mare fuel is exhausted, the star collapses under its own gravity typically forming a white dwarf, with outer layers of mass thrown off as planetary nebula. The white dwarf achieves equilibrium with gravitation collapse now balanced by the outward pressure caused by the electronic matter making up the dwarf being degenerate. The latter implies the density of electronic matter is such that the quantum energy levels are completely occupied by electrons, causing them to stack up in energy owing to the Pauli principle. This leads to a large outward pressure reflecting the difficulty in rearranging particles between energy levels while preventing multiple occupancy. For stars with masses ≳1.5M_(o.)\gtrsim 1.5 M_{\odot}, the white dwarf cannot achieve this equilibrium owing to the size of the inward gravitational force, and a neutron star or black hole can be formed. The former involves electrons and protons fusing to form neutrons, with equilibrium now achieved from the outward pressure of the degenerate neutron matter. Pulsars were discovered by Jocelyn Bell Burnell (1943-) in 1967, and were later identified as rapidly rotating neutron stars Black holes are discussed in Chapter 25. ^(19){ }^{19} We will often invoke the observer at infinity, whose proper time tau=t\tau=t.
To interpret the first equation it can be rewritten as
The first term on the right is the Newtonian prediction ^(18){ }^{18} with the remaining terms giving the relativistic corrections. In low-mass stars, the largest contribution to rho\rho is from baryons (i.e. nuclear matter), which don't contribute significantly to the pressure (which turns out to be provided by electrons). This means p//rho~~0p / \rho \approx 0 and p//m~~0p / m \approx 0, so that the relativistic corrections are not large. In larger stars, the pressure and energy density increase the right-hand side of the equation, steepening the pressure energy density increase the right-hand side of the equation, steepening the pressure the prediction of Newtonian physics
21.4 The meaning of the coordinates
In general relativity, coordinates have no intrinsic metric significance. However, we can relate the coordinates to the quantities of interest in describing stars and orbits in this specific case. This is our next task.
Example 21.4
Let's consider a circle in the equatorial plane (theta=pi//2)(\theta=\pi / 2) at an instant in time ( dt=0\mathrm{d} t=0 ) We then have
Taking a square root and integrating, we find the circumference of the circle is
{:(21.33)C=int_(phi=0)^(2pi)rdphi=2pi r:}\begin{equation*}
C=\int_{\phi=0}^{2 \pi} r \mathrm{~d} \phi=2 \pi r \tag{21.33}
\end{equation*}
We can therefore call rr a radius, in that it supplies the correct scaling to work out the circumference of circles. We must be careful, since if we attempt to compute the distance between two events along a radial line, we obtain
{:(21.34)Delta s=int_(r_(A))^(r_(B))(1-(2M)/(r))^(-(1)/(2))dr:}\begin{equation*}
\Delta s=\int_{r_{\mathrm{A}}}^{r_{\mathrm{B}}}\left(1-\frac{2 M}{r}\right)^{-\frac{1}{2}} \mathrm{~d} r \tag{21.34}
\end{equation*}
which will be greater than r_(B)-r_(A)r_{\mathrm{B}}-r_{\mathrm{A}}.
We conclude that rr does not give the distance from the origin in the Schwarzschild geometry. Since it does give the radius of circles it is sometimes known as the circumferential radial coordinate.
Next, we turn to the time. We find that the proper time between two events is
{:(21.35)Delta tau=(1-(2M)/(r))Delta t:}\begin{equation*}
\Delta \tau=\left(1-\frac{2 M}{r}\right) \Delta t \tag{21.35}
\end{equation*}
For r rarr oor \rightarrow \infty we have Delta tau=Delta t\Delta \tau=\Delta t. We conclude that the tt coordinate is the time between events as measured by a clock at infinity. ^(19){ }^{19}
We now have a suitable metric field we can use it to examine the motion that is possible in the curved spacetime that it describes. In the next chapter, we look at general properties of motion in the Schwarzschild geometry before, in Chapter 23 , turning to the question of orbits.
Chapter summary
The Schwarzschild geometry refers to static, spherically symmetric spacetime.
The metric field in the Schwarzschild geometry is given by the Schwarzschild line element
The Schwarzschild metric is static, asymptotically flat and is badly behaved at r=0r=0 and r=2Mr=2 M.
Birkhoff's theorem says that any spherically symmetric solution to Einstein's equation is identical to the Schwarzschild solution.
Exercises
(21.1) Use the vielbein components to express the components of the energy-momentum tensor T\boldsymbol{T} for a perfect fluid in a (t,r,theta,phi)(t, r, \theta, \phi) coordinate system.
(21.2) (a) Verify eqns 21.10 using the method suggested in the text.
(b) Using eqns 21.25 , write the components of GG in the orthonormal frame using the familiar polar coordinates and show that each component vanishes.
(21.3) Using the symmetries of R\boldsymbol{R}, prove the useful rule in Sidenote 10.
(21.4) (a) Confirm that eqn 21.19 follows from eqn 21.17. (b) Show that eqn 21.19 implies eqn 21.18 in the Newtonian limit.
(21.5) (a) By computing the relevant determinant gg, compute the area of a spherical surface of fixed rr and tt in the Schwarzschild geometry.
(b) Compute a circumference at fixed r,tr, t and theta\theta.
(21.6) (a) Show that in Newtonian gravity, hydrostatic equilibrium requires that
where dm=4pir^(2)rhodr\mathrm{d} m=4 \pi r^{2} \rho \mathrm{~d} r.
Hint: Generalize the usual derivation of Pascal's
law of hydrostatics in a uniform gravitational field for the case of a non-uniform field.
(21.7) Justification of Birkhoff's theorem. The most general spherically symmetric line element has the form
(a) Show that a transformation t rarr t+h(r,t)t \rightarrow t+h(r, t), where h(r,t)h(r, t) is some function, can be used to eliminate the cross term proportional to drdt\mathrm{d} r \mathrm{~d} t, such that we have ds^(2)=-e^(Phi(r,t))dt^(2)+e^(Lambda(r,t))dr^(2)+r^(2)((d)theta^(2)+sin^(2)theta(d)phi^(2))\mathrm{d} s^{2}=-\mathrm{e}^{\Phi(r, t)} \mathrm{d} t^{2}+\mathrm{e}^{\Lambda(r, t)} \mathrm{d} r^{2}+r^{2}\left(\mathrm{~d} \theta^{2}+\sin ^{2} \theta \mathrm{~d} \phi^{2}\right). (21.39)
In Exercise 36.8, we compute the curvature properties of this spacetime. It follows from these that the components of the Einstein tensor are
G_( hat(t) hat(t))=r^(-2)(1-e^(-2Lambda))+2(Lambda,r//r)e^(-2Lambda)G_{\hat{t} \hat{t}}=r^{-2}\left(1-\mathrm{e}^{-2 \Lambda}\right)+2(\Lambda, r / r) \mathrm{e}^{-2 \Lambda}, G_( hat(r) hat(t))=2(Lambda_(t)//r)e^(-(Lambda+Phi))G_{\hat{r} \hat{t}}=2\left(\Lambda_{t} / r\right) \mathrm{e}^{-(\Lambda+\Phi)}, G_( hat(r) hat(r))=2(Phi_(,r)//r)e^(-2Lambda)+r^(-2)(e^(-2Lambda)-1)G_{\hat{r} \hat{r}}=2\left(\Phi_{, r} / r\right) \mathrm{e}^{-2 \Lambda}+r^{-2}\left(\mathrm{e}^{-2 \Lambda}-1\right), G_( hat(theta) hat(theta))=G_( hat(phi) hat(phi)),=G_{\hat{\theta} \hat{\theta}}=G_{\hat{\phi} \hat{\phi}},= (Phi_(,rr)+Phi_(,r)^(2)-Phi_(,r)Lambda_(,r)+Phi_(,r)//r-Lambda_(,r)//r)e^(-2Lambda)\left(\Phi_{, r r}+\Phi_{, r}^{2}-\Phi_{, r} \Lambda_{, r}+\Phi_{, r} / r-\Lambda_{, r} / r\right) \mathrm{e}^{-2 \Lambda} -(Lambda_(,tt)+Lambda_(,t)^(2)-Phi_(,t)Lambda_(,t))e^(-2Phi)-\left(\Lambda_{, t t}+\Lambda_{, t}^{2}-\Phi_{, t} \Lambda_{, t}\right) \mathrm{e}^{-2 \Phi}.
(b) Show that if we are in a vacuum, then Lambda(r,t)\Lambda(r, t) is independent of time.
(c) Use the remaining components of the Einstein equation to show that
where f(t)f(t) is a function of time.
(e) Use this to prove Birkhoff's theorem, which says that the Schwarzschild geometry is the most general, asymptotically flat, spherically symmetric solution of the Einstein equation.
(21.8) Consider filling a universe described by the line element in eqn 21.5 with energy density zeta\zeta, such that we have T_( hat(t) hat(t))=8pi zetaT_{\hat{t} \hat{t}}=8 \pi \zeta and T_( hat(i) hat(i))=-8pi zetaT_{\hat{i} \hat{i}}=-8 \pi \zeta.
(a) Use the results from this chapter to show that this energy density is consistent with Phi=-Lambda\Phi=-\Lambda.
(b) Show that the line element
is a solution to the Einstein equation. Here dOmega^(2)=\mathrm{d} \Omega^{2}=dtheta^(2)+sin^(2)thetadphi^(2)\mathrm{d} \theta^{2}+\sin ^{2} \theta \mathrm{~d} \phi^{2} and HH is a constant you should determine.
This solution is actually the de Sitter model (Universe 1 from Chapter 15) again, expressed in a different set of coordinates. The link is made in Exercise 18.8.
Motion in the Schwarzschild geometry
We are merely the stars' tennis-balls, struck and bandied Which way to please them
John Webster (1580-1625) The Duchess of Malfi
Thanne longen folk to goon on pilgrimages
Geoffrey Chaucer (c.1343-1400)
Prologue to the Cantebury Tales
We found in the last chapter that the Schwarzschild line element is given by
This equation represents the metric field outside a spherically symmetric object, such as a star, of mass MM. In this chapter and the following ones, we shall examine the free-falling motion of particles in this metric field, where the particles follow the geodesics of this geometry. We will find that general relativity allows a richer range of possible motions than are found in the Newtonian problem, including, in addition to the interesting trajectories of massive particles, effects on the motion of photons: the particles of light.
At first glance the problem of relativistic motion looks like a formidable one involving deriving solutions to the geodesic equation, ^(1){ }^{1} with terms provided by the connection coefficients found from eqn 22.2 . However, just as problems involving Newtonian orbits are greatly simplified by knowing about conserved quantities, we shall see how an analysis of conserved quantities, along with simple facts about the velocity u\boldsymbol{u}, allow us to solve a variety of key problems. In particular, we make a lot of use of the identity ^(2)u*u=-1{ }^{2} \boldsymbol{u} \cdot \boldsymbol{u}=-1 for massive particles and u*u=0\boldsymbol{u} \cdot \boldsymbol{u}=0 for massless ones. In addition, just as energy and angular momentum are conserved in the Newtonian potential, we will see that something very much like these is also conserved in the relativistic case, as we might naively expect. We turn first, therefore, to the method for extracting conserved quantities.
22.1 Constants of the motion 238 22.2 Gravitational redshift 239 22.3 Motion in Schwarzschild spacetime 240 22.4 Example: the radial plunge
Chapter summary 244
Exercises 244
In this chapter, we'll need to use the components of the Schwarzschild metric
with the connection coefficients determined from the metric. ^(2){ }^{2} Reminder: A massive particle's world line is parametrized by proper time so that, in a coordinate system with x^(mu)=x^{\mu}= ( t,r,theta,phit, r, \theta, \phi ), the velocity u\boldsymbol{u} has components u^(mu)=(u^(t),u^(r),u^(theta),u^(phi))u^{\mu}=\left(u^{t}, u^{r}, u^{\theta}, u^{\phi}\right), i.e.
↷\curvearrowright Killing vectors are discussed in detail in Chapter 33 in the context of geometry.
22.1 Constants of the motion
One of the most important factors in analysing motion in mechanics is the identification of constants of the motion. In geometrical problems, exemplified by the physics of gravitation, conserved quantities can be identified by finding a set of fields known as the Killing vector fields xi\boldsymbol{\xi} of the metric. These are very simply identified by inspection, by noting which coordinates do not feature in the components of the metric. For now we will give (but not prove) a simple recipe for identifying conserved quantities.
To find a conserved quantity from a metric field:
Step 1: Look at the metric components and identify those coordinates on which none of the metric components depend.
Step 2: If the variable x^(alpha)x^{\alpha} does not feature, we have a Killing vector xi=e_(alpha)\boldsymbol{\xi}=\boldsymbol{e}_{\alpha}.
Step 3: Particles travelling along a geodesic have a velocity u=u^(mu)e_(mu)\boldsymbol{u}=u^{\mu} \boldsymbol{e}_{\mu}, which is the tangent to a geodesic. The quantity
is conserved along the geodesic.
To identify the Killing vectors for the Schwarzschild geometry, we spot (step 1) that the metric components are all independent of the coordinates tt and phi\phi. Ordering the coordinates (t,r,theta,phi)(t, r, \theta, \phi) we identify two corresponding Killing vectors (step 2). First, from the independence of tt we extract a Killing vector xi\boldsymbol{\xi} via
{:(22.4)xi=e_(t)","quadxi^(mu)=(1","0","0","0)","quad(" time independence "):}\begin{equation*}
\boldsymbol{\xi}=\boldsymbol{e}_{t}, \quad \xi^{\mu}=(1,0,0,0), \quad(\text { time independence }) \tag{22.4}
\end{equation*}
and from the independence of phi\phi we identify a Killing vector eta\boldsymbol{\eta} as we have
We now use the fact that xi*u\boldsymbol{\xi} \cdot \boldsymbol{u} is a constant along a geodesic (step 3) to identify the conserved quantities. The first is identified as^(3)\mathrm{as}^{3}
This quantity tilde(E)\tilde{E} tells us about the energy per unit rest mass. ^(4){ }^{4} In general, it can be interpreted, for timelike geodesics, as total energy per unit rest mass, measured by a static observer at infinity.
The other conserved quantity, called tilde(L)\tilde{L} is also straightforward to pick out as eta*u=u_(phi)\boldsymbol{\eta} \cdot \boldsymbol{u}=u_{\phi}, or, from u_(phi)=g_(phi phi)u^(phi)u_{\phi}=g_{\phi \phi} u^{\phi}
We interpret this as angular momentum per unit rest mass. Note that for equatorial motion (i.e. with theta=pi//2\theta=\pi / 2 ) this becomes tilde(L)=r^(2)phi^(˙)\tilde{L}=r^{2} \dot{\phi}, which looks just like the expression for angular momentum per unit mass in cylindrical coordinates.
So far we have considered only the geometry, and not what sort of particle is in free fall. This information is supplied via the square of the velocity vector u*u\boldsymbol{u} \cdot \boldsymbol{u}. Massive particles move along timelike geodesics and so we fix the magnitude of the velocity vector (which is tangent to the geodesic) by saying
where the proper time tau\tau has been employed as an affine parameter marking off the trajectory of the particle. Photons travel along null geodesics, whose velocity tangent vector has the property
where lambda\lambda is an affine parameter for the photon world line.
22.2 Gravitational redshift
With these preliminaries, we can derive some results. The first is a result for photons: the gravitational redshift caused by the Schwarzschild geometry. ^(5){ }^{5} To analyse this, we consider the motion of a photon with null momentum p\boldsymbol{p} propagating along a radial line. By symmetry, this is a geodesic. The photon has the property ^(6){ }^{6} that p*xi\boldsymbol{p} \cdot \boldsymbol{\xi} is conserved along the geodesic, which implies that, in this geometry, g_(tt)p^(t)=(1-2M//r)p^(t)(r)g_{t t} p^{t}=(1-2 M / r) p^{t}(r) is conserved as rr is varied (rather than the flat-space energy p^(t)p^{t}, as we might naively have expected). Remember, however, that an observer at rr does not measure p^(t)p^{t}; they measure the local value p^( hat(t))=ℏomegap^{\hat{t}}=\hbar \omega.
We can find out how the photon's measured frequency omega\omega changes as the photon moves radially outwards from a massive star. We consider two static observers, one at radius r=Rr=R and one at r=oor=\infty (Fig. 22.1). Being observers, their world lines have tangents that are timelike velocity vectors. The energy of a photon with momentum p\boldsymbol{p}, measured by an observer ^(7){ }^{7} with velocity u_("obs ")\boldsymbol{u}_{\text {obs }} is
If the observer is at rest with respect to the star, then u_("obs ")^(i)=0u_{\text {obs }}^{i}=0 and we can determine the timelike component of u_("obs ")\boldsymbol{u}_{\text {obs }} using the velocity identity
Fig. 22.1 A photon world line meets an observer at RR and one at infinity. ^(5){ }^{5} Here we revisit our second cause of frequency shifts discussed earlier in Chapter 6. (The others are the Doppler effect and the cosmological redshift.) ^(6){ }^{6} This follows since we shall assume p\boldsymbol{p} to be parallel to u\boldsymbol{u} for light and p\boldsymbol{p} is therefore tangent to the geodesics. (For a more careful discussion of photon momentum, see Section 24.7.) ^(7){ }^{7} An alternative rule would be to use the vielbein components to say that the measured energy p^(t)p^{t} is related to p^(t)p^{t} via
This alternative approach is examined in the exercises. Here, instead, since we are trying to relate measurements made by local observers in two places, we use the conservation laws encoded in the Killing vector xi\boldsymbol{\xi} (i.e. that u*xi\boldsymbol{u} \cdot \boldsymbol{\xi} is conserved along a geodesic). Our strategy is therefore to incorporate the energy Killing vector xi\boldsymbol{\xi} into the description of the stationary observer's velocity.
With the Schwarzschild metric component g_(tt)g_{t t}, this gives a velocity com- ^(8){ }^{8} Recall that xi\xi has components xi^(mu)=\xi^{\mu}= ( 1,0,0,01,0,0,0 ).
Fig. 22.2 A plot of omega_(R)\omega_{R} against RR for R > 2MR>2 M.
We can summarize the important results that we shall use to compute trajectories:
For massive particles, the particle velocity u\boldsymbol{u} is tangent to the world line and has the property u*u=-1\boldsymbol{u} \cdot \boldsymbol{u}=-1.
■ For photons we have u*u=0\boldsymbol{u} \cdot \boldsymbol{u}=0. The interval ds=0\mathrm{d} s=0 on the world line provides a useful constraint.
We compute conservation laws using the Killing rule, that says that if the metric components are independent of variable x^(alpha)x^{\alpha}, the component u_(alpha)u_{\alpha} is conserved along a geodesic.
To access the coordinate velocity dx^(i)//dt\mathrm{d} x^{i} / \mathrm{d} t we use the trick
^(9){ }^{9} The approach described here is covered in many books, but beware that conventions for defining some of the terms differ. We follow the approach and notation used in Hartle.
ponent at a radius rr of
or in terms of one of the Killing vectors, ^(8)u_("obs ")(r)=(1-2M//r)^(-(1)/(2))xi{ }^{8} \boldsymbol{u}_{\text {obs }}(r)=(1-2 M / r)^{-\frac{1}{2}} \boldsymbol{\xi}, which is the key equation. This expression means we can write eqn 22.12 in terms of the Killing vector xi\boldsymbol{\xi}. Specifically, the energy measured by the stationary observer sat at a radius RR is
where subscript RR implies the quantity is evaluated at this radius. For the energy measured by the observer at infinity, the factor (1-2M//r)^(-(1)/(2))rarr1(1-2 M / r)^{-\frac{1}{2}} \rightarrow 1 and so ℏomega_(oo)=(-xi*p)_(oo)\hbar \omega_{\infty}=(-\boldsymbol{\xi} \cdot \boldsymbol{p})_{\infty}. However, since the photon is moving along a geodesic, the quantity xi*p\boldsymbol{\xi} \cdot \boldsymbol{p} is conserved in the motion, and so (-xi*p)_(R)=(-xi*p)_(oo)(-\boldsymbol{\xi} \cdot \boldsymbol{p})_{R}=(-\boldsymbol{\xi} \cdot \boldsymbol{p})_{\infty} and we must therefore have
The frequency at infinity is less than the frequency at the point RR (see Fig. 22.2). One can rationalize this result by saying that the photon's energy is lowered through its climbing of the gravitational potential.
22.3 Motion in Schwarzschild spacetime
We now turn to the motion of massive particles in the Schwarzschild geometry. ^(9){ }^{9} It might be assumed that we need to solve the full geodesic equation to understand the motion of particles and photons. Fortunately for us, the only ingredients needed are the constants of the motion given by the Killing vectors, and the velocity identity u*u=-1\boldsymbol{u} \cdot \boldsymbol{u}=-1.
Example 22.1
For simplicity, we assume motion in the equatorial plane of the geometry, which is to say that we fix theta=pi//2\theta=\pi / 2 and so u^(theta)=0u^{\theta}=0. To analyse motion of a particle we start by using the velocity identity for the particle u*u=g_(mu nu)u^(mu)u^(nu)=-1\boldsymbol{u} \cdot \boldsymbol{u}=g_{\mu \nu} u^{\mu} u^{\nu}=-1, to write
Now we introduce a new constant energy-like variable, ^(10)E=( tilde(E)^(2)-1)//2{ }^{10} \mathcal{E}=\left(\tilde{E}^{2}-1\right) / 2, and define an effective potential for motion in the Schwarzschild geometry of
and we end up with the familiar equation E=(1)/(2)(((d)r)/((d)tau))^(2)+V_("eff ")(r)\mathcal{E}=\frac{1}{2}\left(\frac{\mathrm{~d} r}{\mathrm{~d} \tau}\right)^{2}+V_{\text {eff }}(r).
The result of the last example is that the motion of a massive particle obeys the effective-potential equation
This is similar to the equation for motion in an effective potential that we saw ^(11){ }^{11} in Chapter 20 for Newtonian motion. We conclude that motion takes place in an effective potential V_("eff ")V_{\text {eff }}. The potential for the Schwarzschild geometry is shown in Fig. 22.3. Compared to the Newtonian potential, this one has more structure: specifically, with increasing rr, an initial increase in V_("eff ")V_{\text {eff }} to a maximum, followed by behaviour that looks similar to its Newtonian cousin. This new structure leads to the richness of the new trajectories allowed by relativity.
Example 22.2
We can restore factors of cc and GG to find
{:(22.26)E_(N)=(m)/(2)(((d)r)/((d)tau))^(2)+(L^(2))/(2mr^(2))-(GMm)/(r)-(GML^(2))/(mc^(2)r^(3)):}\begin{equation*}
E_{\mathrm{N}}=\frac{m}{2}\left(\frac{\mathrm{~d} r}{\mathrm{~d} \tau}\right)^{2}+\frac{L^{2}}{2 m r^{2}}-\frac{G M m}{r}-\frac{G M L^{2}}{m c^{2} r^{3}} \tag{22.26}
\end{equation*}
where L=m tilde(L)L=m \tilde{L}. We conclude that the usual Newtonian energy equation is augmented at order 1//c^(2)1 / c^{2} by the relativistic, 1//r^(3)1 / r^{3} term.
We are now in the position to be able to provide a description of the motion. If we are interested in what would be measured by an observer, we will need access to the vielbein components for the Schwarzschild geometry in order to shift into the orthonormal frame of the observer. The vielbein for a stationary observer is given in the margin. It is worth noting that for this observer at infinity, measurements of the hat(t)\hat{t} and hat(r)\hat{r} components of a vector will give the coordinate values as both of the relevant vielbein components are unity at infinity. ^(10){ }^{10} Particles coming in from infinity have tilde(E) > 1\tilde{E}>1 and so have effective energy E >\mathcal{E}> E.
0. ^(11){ }^{11} The equations of motion differ by a factor of mm. The Newtonian version has factor of mm. The Newtonian version has
an effective potential energy U_("eff ")U_{\text {eff }} and so an effective potential
an effective potential
The relativistic version has an extra term with a 1//r^(3)1 / r^{3} dependence.
Fig. 22.3 The relativistic effective potential for a particle (solid line, with tilde(L)//M=4.3\tilde{L} / M=4.3 ), compared to the Newtonian one (dotted line).
The vielbein components for an observer in the orthonormal frame (where the observer is stationary with respect to the coordinate frame) has components
It is important to remember that if we want to shift into the frame of a moving observer we will need a different vielbein. An example we have seen before that we will use several times is the freely falling observer's frame.
Fig. 22.4 The radial plunge. ^(12){ }^{12} At infinity, spacetime is flat and the particle is at rest. Initially, the components of the velocity must therefore be u^(mu)(r=oo)=(dx^(mu))/(dtau)=(1,0,0,0)u^{\mu}(r=\infty)=\frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau}=(1,0,0,0). Recall also that the energy
^(13){ }^{13} There is no mechanism to pick up angular velocity, hence the zeros for the theta\theta and phi\phi components of velocity. ^(14){ }^{14} The falling observer's local frame is flat, but moving with respect to the coordinate frame, so differs from the conventional orthonormal frame we often use. To find this frame, note that we always take e_( hat(t))=u\boldsymbol{e}_{\hat{t}}=\boldsymbol{u}. Then, for convenience choose (e_( hat(theta)))^(mu)=(0,0,1,0)\left(\boldsymbol{e}_{\hat{\theta}}\right)^{\mu}=(0,0,1,0) and (e_( hat(phi)))^(mu)=(0,0,0,1)\left(\boldsymbol{e}_{\hat{\phi}}\right)^{\mu}=(0,0,0,1). This means that the one remaining vielbein component is
{:(22.30)(e_( hat(r)))^(mu)=([(-sqrt(2M//r))/(1-2M//r)],[1],[0],[0]):}\left(\boldsymbol{e}_{\hat{r}}\right)^{\mu}=\left(\begin{array}{c}
\frac{-\sqrt{2 M / r}}{1-2 M / r} \tag{22.30}\\
1 \\
0 \\
0
\end{array}\right)
^(15){ }^{15} In this section, it is necessary to fix the value of rr at some value of the proper time tau\tau in order to fix the integration constant. ^(16){ }^{16} A useful quantity to note for computations is the coordinate velocity
The magnitude of this quantity is the coordinate escape velocity of a particle, as discussed in the exercises.
22.4 Example: the radial plunge
Let's consider a massive particle that starts at rest at infinity and plunges towards a star along a radial line (Fig. 22.4). This line is a geodesic.
Example 22.3
We have immediately that the angular momentum tilde(L)=0\tilde{L}=0 since the plunge is radial. 'At rest at infinity' implies that at a great distance from the gravitating object we have ^(12)u^(t)=dt//dtau=1{ }^{12} u^{t}=\mathrm{d} t / \mathrm{d} \tau=1 and so, for the duration of the freefall, tilde(E)=1\tilde{E}=1 and E=0\mathcal{E}=0. The fact that tilde(E)=1\tilde{E}=1 allows us to pick out, at later times, that
which provides us with an expression for the radial velocity u^(r)=dr//dtau=u^{r}=\mathrm{d} r / \mathrm{d} \tau=-(2M//r)^((1)/(2))-(2 M / r)^{\frac{1}{2}}, where the sign is chosen so that the particle is falling towards the gravitating star. We conclude that the velocity during the fall is given by a vector u\boldsymbol{u} with components in the coordinate frame of ^(13){ }^{13}
where the negative square root has been chosen for an inward-travelling particle and tau_(**)\tau_{*} fixes the value of the proper time ^(15){ }^{15} at the end of the plunge, when r=0r=0. This can be integrated with the result that
The variation of tau\tau with the Schwarzschild radius coordinate rr is shown in Fig. 22.5. Notice that, from an arbitrary starting value of rr it takes a finite amount of proper time to reach any value of rr, as we might expect.
Next, the amount of elapsed coordinate time tt. Since (e_(t))^( bar(t))=1\left(e_{t}\right)^{\bar{t}}=1 at infinity, this is also the time measured by an observer at infinity watching the particle plunge. We have ^(16){ }^{16}
where t_(**)t_{*} gives the time coordinate when r=0r=0. This function is graphed in Fig. 22.6.
There are some curious facts about the results from the last example: the most notable being that, from the point of view of the observer at infinity, it takes an infinite amount of coordinate time for an in-falling astronaut to reach r=2Mr=2 M (but a finite proper time). We will return to this point in Chapter 25.
Finally, let's examine the coordinate velocity of the particle that plunges radially in a Schwarzschild metric.
Example 22.5
We won't assume the particle starts from rest this time. The velocity identity gives us
where, in the final line we've written the equation in terms of the constant of the motion u*xi=u_(t)\boldsymbol{u} \cdot \boldsymbol{\xi}=u_{t}. Solving for (dr//dt)^(2)(\mathrm{d} r / \mathrm{d} t)^{2} we find
This coordinate velocity vanishes as the particle approaches r=2Mr=2 M (Fig. 22.7).
What does a stationary observer determine as the coordinate velocity? We won't assume that they are at infinity here and so we shift to the orthonormal frame using the vielbein. Remember that for 1-forms like dt\boldsymbol{d} t and dr\boldsymbol{d} r we have omega^( hat(mu))=(e_(mu))^( hat(mu))omega^(mu)\boldsymbol{\omega}^{\hat{\mu}}=\left(\boldsymbol{e}_{\mu}\right)^{\hat{\mu}} \boldsymbol{\omega}^{\mu}. We find
{:[d hat(t)=(1-(2M)/(r))^((1)/(2))dt],[(22.39)d hat(r)=(1-(2M)/(r))^(-(1)/(2))dr.]:}\begin{align*}
& \boldsymbol{d} \hat{t}=\left(1-\frac{2 M}{r}\right)^{\frac{1}{2}} \boldsymbol{d} t \\
& \boldsymbol{d} \hat{r}=\left(1-\frac{2 M}{r}\right)^{-\frac{1}{2}} \boldsymbol{d} r . \tag{22.39}
\end{align*}
Noting that the derivative dr//dt-=dr//dt\mathrm{d} r / \mathrm{d} t \equiv \boldsymbol{d} r / \boldsymbol{d} t, we have that the stationary observer observes a plunge with a velocity
Irrespective of u_(t)u_{t}, and hence the initial velocity, this locally measured velocity approaches unity or, in real units the speed of light, as r rarr2Mr \rightarrow 2 M (Fig. 22.7).
Using only the velocity identity and the conservation of energy and angular momentum, we have shown a number of curious relativistic effects in a particle's motion. We'll pick up these points in a few chapters' time. In the next chapter, we turn to orbits.
Fig. 22.5 The evolution of the coordinate rr with the proper time tau\tau for the radial plunge.
Fig. 22.6 The evolution of the coordinate rr with the coordinate time tt for the radial plunge. The dotted line shows r_(S)=2Mr_{\mathrm{S}}=2 M.
Fig. 22.7 Example evolution of the coordinate velocity v=dr//dtv=\mathrm{d} r / \mathrm{d} t with radial coordinate rr for the radial plunge (solid line). The dashed line shows the coordinate velocity hat(v)=d hat(r)//d hat(t)\hat{v}=\mathrm{d} \hat{r} / \mathrm{d} \hat{t} measured by a stationary observer.
Chapter summary
Motion in the Schwarzschild geometry can often be computed using the velocity u\boldsymbol{u}, subject to the constraint that u*u=-1\boldsymbol{u} \cdot \boldsymbol{u}=-1. The constants of the motion tilde(E)\tilde{E} and tilde(L)\tilde{L} follow from the independence of the metric components from coordinates tt and phi\phi.
The effective potential method allows motion to be analysed in terms of a relativistic V_("eff ")V_{\text {eff }}.
A particle that plunges from rest at infinity has E=0\mathcal{E}=0 and a velocity u\boldsymbol{u} with components
22.1) Consider the acceleration a=grad_(u)u\boldsymbol{a}=\boldsymbol{\nabla}_{\boldsymbol{u}} \boldsymbol{u}. By the geodesic equation, the components of the acceleration a^(mu)=u^(alpha)u^(mu)_(;alpha)a^{\mu}=u^{\alpha} u^{\mu}{ }_{; \alpha} vanish in free fall.
(a) What are the components of the acceleration for a particle at rest in a static gravitational field with metric components g_(mu nu)g_{\mu \nu} ?
(b) What are the components of the acceleration of a particle at rest at a fixed value of rr, in a Schwarzschild coordinate system.
(c) What is the proper acceleration alpha\alpha (i.e. the acceleration measured in the frame of the particle)?
(22.2) Although we have used a variety of shortcuts to compute the equations of motion, the same information is available from the geodesic equation.
(a) Compute the components of the geodesic equation for the Schwarzschild geometry with line element given in the form of eqn 22.2 .
If you've done this previously for the line element expressed in different variables, you could simply re-express the equations in terms of (t,r,theta,phi)(t, r, \theta, \phi).
(b) How do these expressions simplify for motion in the plane with theta=pi//2\theta=\pi / 2 ?
(c) Show that the equation of motion for the rr coordinate is consistent with eqn 22.22 .
(22.3) Derive eqn 22.16 using the vielbein components, and the fact that p_(t)p_{t} is conserved on the geodesic.
(22.4) Verify eqn 22.26 in the Newtonian limit.
22.5) A stationary observer on the surface of a spherical mass MM of radius RR launches a projectile at its escape velocity. It does not experience any force during its motion and so it follows a geodesic.
(a) What is the escape velocity of the projectile measured in the observer's rest frame?
(b) What is the energy of the projectile as measured by the observer?
(22.6) Gravitational clock effect: Consider the Schwarzschild geometry describing the gravitational field of the Earth, which is rotating on its axis with angular velocity omega\omega.
(a) For an observer at rest on the Earth's surface at the equator, show that the proper time interval is given approximately by
{:(22.42)dtau_(1)~~(1-(M)/(r)-(r^(2)omega^(2))/(2))dt:}\begin{equation*}
\mathrm{d} \tau_{1} \approx\left(1-\frac{M}{r}-\frac{r^{2} \omega^{2}}{2}\right) \mathrm{d} t \tag{22.42}
\end{equation*}
(b) Now consider a second observer who flies eastwards around the equator with velocity vv at a height hh, whose measures a proper time interval dtau_(2)\mathrm{d} \tau_{2}. Show that
{:(22.43)dtau_(1)-dtau_(2)~~[-(Mh)/(r^(2))+((2r omega+v)v)/(2)]dt:}\begin{equation*}
\mathrm{d} \tau_{1}-\mathrm{d} \tau_{2} \approx\left[-\frac{M h}{r^{2}}+\frac{(2 r \omega+v) v}{2}\right] \mathrm{d} t \tag{22.43}
\end{equation*}
(c) Define Delta=(dtau_(1)-dtau_(2))//dtau_(1)\Delta=\left(\mathrm{d} \tau_{1}-\mathrm{d} \tau_{2}\right) / \mathrm{d} \tau_{1} and estimate this quantity for an eastward flight around the Earth.
(d) Constrast the value of Delta\Delta for a similar flight in a westward direction.
This experiment was carried out in the early 1970s by Hafele and Keating, whose results were consistent with predictions in this problem fwhich follows the method in Ryder (2009)].
(22.7) Two observers start on the same radial line in the Schwarzschild geometry, one at rest at r_(1)r_{1} and an-
other at rest at r_(2) > r_(1)r_{2}>r_{1}. At t=0t=0, the observer at r_(2)r_{2} begins to fall freely. What is the relative velocity of the observers as they meet?
Hint: Recall from Chapter 2 that gamma\gamma, corresponding to the instantaneous relative velocity v_("rel ")v_{\text {rel }}, can be found by taking the dot product of the two observers' velocity vectors: -gamma(v_("rel "))=u*v-\gamma\left(v_{\text {rel }}\right)=\boldsymbol{u} \cdot \boldsymbol{v}.
23
Orbits in the Schwarzschild geometry
23.1 Orbits for massive particles
23.2 Stable circular orbits 248 23.3 Precession of the perihelion
Chapter summary 252
and tilde(L)=r^(2)sin^(2)thetau^(phi)\tilde{L}=r^{2} \sin ^{2} \theta u^{\phi}. ^(1){ }^{1} See Exercise 22.2.
I am about to make my last voyage, a great leap in the dark Thomas Hobbes (1588-1679)
The description of orbits given in Chapter 20 proved a crowning triumph of Newtonian mechanics. One of the first successes of general relativity was to provide predictions of orbits that, although similar to Kepler and Newton's ellipses, actually provided closer agreement with observation. We now turn to the possible orbits in the Schwarzschild geometry. These are a special class of geodesic, where a particle executes periodic motion by freely falling along a closed path in space. As in the last chapter, we will make use of the Schwarzschild line element and the conserved quantities EE and LL. Previously, we have confined our attention to those plunging motions with L=0L=0. We now relax that constraint in order to allow angular motion. Our discussion of the resulting stable orbits will yield the famous prediction of the precession of the perihelion of an orbit in the Schwarzschild geometry.
23.1 Orbits for massive particles
Orbits made by massive particles are on geodesics parametrized by their proper time tau\tau. For simplicity, we restrict our attention to particles moving in an equatorial plane, so that we can set theta=pi//2\theta=\pi / 2 and so u^(theta)=u^{\theta}=dtheta//dtau=0\mathrm{d} \theta / \mathrm{d} \tau=0. There is no loss of generality here since, as in the Newtonian case, orbiting particles are confined to a plane, as we shall now prove.
Example 23.1
If we consider the geodesic equation found by varying the Schwarzschild line element with respect to the angle ^(1)theta{ }^{1} \theta, we obtain
This equation of motion is solved by theta=pi//2\theta=\pi / 2 for all tau\tau. At this angle the right-hand side of the equation is zero and so the quantity r^(2)dtheta//dtaur^{2} \mathrm{~d} \theta / \mathrm{d} \tau is conserved. Since we initially fix (d)/(d tau)theta=0\frac{d}{d \tau} \theta=0, the particle never acquires any acceleration, and the motion is confined to the plane. Just as in the Newtonian case, the confinement to the plane can be attributed to conservation of angular momentum. That is, if we rotate our coordinates such that we initially have theta=(pi)/(2)\theta=\frac{\pi}{2}, then the equation of motion for u^(phi)u^{\phi} gives (d)/(dtau) tilde(L)=0\frac{\mathrm{d}}{\mathrm{d} \tau} \tilde{L}=0 on the whole geodesic.
Our strategy will be to understand the possible orbits by analysing an effective potential, much as we did both in the Newtonian case and also in the last chapter. Using the effective energy variable E=( tilde(E)^(2)-1)//2\mathcal{E}=\left(\tilde{E}^{2}-1\right) / 2,
As we noted in the last chapter, this setup is designed to look as much like the Newtonian version as possible. ^(2){ }^{2} Here, the extra attractive term -M tilde(L)^(2)//r^(3)-M \tilde{L}^{2} / r^{3} is the source of the richness of relativistic trajectories. This term is largest for large MM and tilde(L)\tilde{L} and small values of rr.
An example of the effective potential V_("eff ")V_{\text {eff }} is shown in Fig. 23.1, where we see its two characteristic extrema: (i) a minimum at a radius r_(+)r_{+}, which resembles the minimum in the Newtonian effective potential; and (ii) a maximum is a smaller radius r_(-)r_{-}, which has no analogue in Newtonian physics. These turning points in the relativistic potential, occurring for V_("eff ")^(')(r_(+-))=0V_{\text {eff }}^{\prime}\left(r_{ \pm}\right)=0, are found at
This equation implies that the nature of the extrema depends on the ratio tilde(L)//M\tilde{L} / M. That is, a combination of the angular momentum of the particle in motion and the mass of the gravitating object determines the effective potential, just as in Newtonian physics.
Example 23.2
In Fig. 23.2, we show some examples of motion in the potential determined by one choice of MM and (non-zero) tilde(L)\tilde{L}. In these plots, we represent V_("eff ")(r)V_{\text {eff }}(r) and the particle's value of E\mathcal{E}. Where E=V_("eff ")\mathcal{E}=V_{\text {eff }} and the lines meet, the particle has no velocity in the radial direction, although it is important to note that the particle has a non-zero value of tilde(L)\tilde{L} (and therefore u^(phi)u^{\phi} ) and so is still in motion.
Figure 23.2(a) shows a circular orbit, with a particle sat at the minimum of V_("eff ")V_{\text {eff }} at a (constant) radius r_(+)r_{+}. The particle has effective energy E=V_("eff ")(r_(+))\mathcal{E}=V_{\text {eff }}\left(r_{+}\right), which takes a negative value, corresponding to the particle being bound in a potential.
Figure 23.2(b) shows a particle in-falling from infinity (so therefore having a positive effective energy ^(3)E{ }^{3} \mathcal{E} ), but colliding with the peak in the potential, which scatters it back off to infinity.
Figure 23.2(c) shows the motion of a particle with a still higher energy. This one - Figure 23.2 (c) shows the motion of a particle with a still higher energy. This one
is not scattered since its effective energy is too high for it to hit the peak in the potential V_("eff ")(r_(-))V_{\text {eff }}\left(r_{-}\right). Instead, this particle spirals towards the centre of the gravitating mass, where it presumable crashes into the mass distribution at some small radius. This trajectory is a consequence of the relativistic potential being distinct from the Newtonian version, in that the latter has an energetic barrier at small rr. This means that a Newtonian particle with positive energy, as long as it has some angular momentum, will always be deflected by the potential (assuming it doesn't crash into the source of the potential first!). ^(2){ }^{2} Recall the Newtonian effective potential energy is given by
U_(eff)(r)=(L^(2))/(2mr^(2))-(GMm)/(r)U_{\mathrm{eff}}(r)=\frac{L^{2}}{2 m r^{2}}-\frac{G M m}{r}
Fig. 23.1 The relativistic effective potential for given values of tilde(L)//M\tilde{L} / M.
(a) V_("eff ")( vec(r))V_{\text {eff }}(\vec{r})
(c) V_("eff ")( vec(r))V_{\text {eff }}(\vec{r})
Fig. 23.2 Some allowed trajectories in the relativistic potential (computed using eqn 23.15). (a) Circular; (b) an encounter with the star leading to scattering; (c) a spiral into the star. ^(3){ }^{3} Recall that particles arriving from infinity have tilde(E) > 1\tilde{E}>1 so E=( tilde(E)^(2)-1)//2 >\mathcal{E}=\left(\tilde{E}^{2}-1\right) / 2> ^(4){ }^{4} These particles will simply spiral into the origin. ^(5){ }^{5} In addition to the minimum, it's notable that the maximum of V_("eff ")V_{\text {eff }} also represents a possible circular orbit. Being a maximum, it is manifestly unstable. The analogous trajectory will be considered for photons in the next chapter.
One thing that we can spot from eqn 23.4 and Fig. 23.1 is that if tilde(L)//M <\tilde{L} / M<sqrt12(=3.464 dots)\sqrt{12}(=3.464 \ldots) then there is only one extremum, a point of inflection. For the corresponding motions at low L//ML / M there is not enough angular momentum to stabilize an orbit. The stable orbits are only possible for tilde(L)//M >= sqrt12\tilde{L} / M \geq \sqrt{12} and, as we shall discover below, these stable orbits are almost, but not quite, elliptical. The difference between these orbits and their Newtonian analogues is that the motion can precess, such that the near-ellipses traced by the trajectories rotate.
It is also the case that if tilde(L)//M=4\tilde{L} / M=4, then the maximum of V_("eff ")V_{\text {eff }} occurs at r_(-)=4Mr_{-}=4 M where V_("eff ")=0V_{\text {eff }}=0, which is significant as it means that particles coming in from infinity, which have E > 0\mathcal{E}>0, can never be scattered by such a potential. ^(4){ }^{4} If the angular momentum is greater than 4M4 M then there are possible trajectories that can scatter particles coming in from infinity. Less angular momentum than this implies a spiral into the origin for E > 0\mathcal{E}>0 particles.
23.2 Stable circular orbits
In the Newtonian case, we saw how the minimum of the effective potential gave us the circular orbits. We can repeat the computation for the relativistic case and show the properties of circular orbits here too. ^(5){ }^{5} Stable circular orbits occur when r=r_(+)r=r_{+}in eqn 23.4. These depend on tilde(L)//M\tilde{L} / M, with the minimum angular momentum occurring for tilde(L)//M=sqrt12\tilde{L} / M=\sqrt{12}, for which we have r_(+)=6Mr_{+}=6 \mathrm{M}. This represents a minimum radius for stable circular orbits in the Schwarzschild geometry.
Example 23.3
From Chapter 20 we know that for a circular orbit we have that the effective potential is a minimum and that the effective energy E=( tilde(E)^(2)-1)//2\mathcal{E}=\left(\tilde{E}^{2}-1\right) / 2 equals the value of the effective potential V_("eff ")(r_(+))V_{\text {eff }}\left(r_{+}\right). Rearranging eqn 23.2 , this amounts to the condition
This equation, along with eqn 23.4 , can be usefully combined to find the ratio tilde(L)// tilde(E)\tilde{L} / \tilde{E} for a circular orbit. From eqn 23.4 we have
{:(23.6)(2Mr_(+)- tilde(L)^(2))^(2)= tilde(L)^(4)(1-(2M^(2))/( tilde(L)^(2))):}\begin{equation*}
\left(2 M r_{+}-\tilde{L}^{2}\right)^{2}=\tilde{L}^{4}\left(1-\frac{2 M^{2}}{\tilde{L}^{2}}\right) \tag{23.6}
\end{equation*}
This is, of course, the same as Kepler's original version of his law, and so we conclude that Kepler's third law for circular orbits is not altered by relativity.
23.3 Precession of the perihelion
The power of general relativity lies partially in explaining effects we find in observations that could not occur in Newtonian gravity. Historically, one of the most important was the precession of the perihelion of mercury. This is one of the three classical solar-system tests of general relativity. Below we shall derive the equation for the trajectory of a particle in the relativistic effective potential. The general idea is illustrated in the following example.
Example 23.4
The precessing orbit is shown in Fig. 23.3 (for a vastly exaggerated angle of precession compared to what we will compute below). The particle oscillates between two points on the effective potential curve, hitting maximum and minimum distances from the origin. As it does so, the trajectories resemble ellipses whose axes precess in the plane of motion.
Since our trajectories are confined to the plane theta=pi//2\theta=\pi / 2, then in order to find the shape of planetary orbits we must write an equation for the coordinate rr in terms of phi\phi (or vice-versa). Since we are not solving the full geodesic equations, but instead exploiting (i) the square of u\boldsymbol{u} and (ii) the conserved quantities tilde(E)\tilde{E} and tilde(L)\tilde{L}, the way to do this is to find dr//dtau\mathrm{d} r / \mathrm{d} \tau and dphi//dtau\mathrm{d} \phi / \mathrm{d} \tau and take their ratio to find dr//dphi\mathrm{d} r / \mathrm{d} \phi.
Starting with the velocity identity u*u=-1\boldsymbol{u} \cdot \boldsymbol{u}=-1, for fixed theta\theta we have
Noting that g^(rr)(u_(r))^(2)=g_(rr)(u^(r))^(2)g^{r r}\left(u_{r}\right)^{2}=g_{r r}\left(u^{r}\right)^{2} for this diagonal metric, ^(7){ }^{7} we have
The three classical solar system tests of relativity are:
Precession of the perihelion of mercury;
Bending of light by the sun;
Gravitational redshift.
We met gravitational redshift in Chapter 6 . In Chapter 9, we also met a fourth test: Shapiro time delay.
Fig. 23.3 The precessing orbit in the relativistic potential calculated using eqn 23.15. The motion is bounded at the points shown on the left, represented by the dotted lines on the right.
The components of the inverse Schwarzschild metric are
^(7){ }^{7} That is, g^(rr)(u_(r))^(2)=g^(rr)(g_(rr)u^(r))^(2)g^{r r}\left(u_{r}\right)^{2}=g^{r r}\left(g_{r r} u^{r}\right)^{2} and g_(rr)g^(rr)=1g_{r r} g^{r r}=1.
and so
Since we also know from u^(phi)=g^(phi phi) tilde(L)u^{\phi}=g^{\phi \phi} \tilde{L} that (dphi)/(dtau)=(( tilde(L)))/(r^(2))\frac{\mathrm{d} \phi}{\mathrm{d} \tau}=\frac{\tilde{L}}{r^{2}}, we are able to isolate the equation that gives us the trajectories in the Schwarzschild geometry through the multiplication (dr)/((d)tau)((d)tau)/((d)phi)\frac{\mathrm{d} r}{\mathrm{~d} \tau} \frac{\mathrm{~d} \tau}{\mathrm{~d} \phi}, which gives
This is our basic equation that tells us the paths in Schwarzschild coordinates that particles can take. ^(8){ }^{8} We can analyse this equation using some of the same techniques that we used in Chapter 20, where we introduced the variable ^(9)u=1//r{ }^{9} u=1 / r.
tion is (dr)/((d)theta)=\frac{\mathrm{d} r}{\mathrm{~d} \theta}=
+-(r^(2))/(L){2m[E+(GMm)/(r)]-(L^(2))/(r^(2))}_((23.16))^((1)/(2))\pm \frac{r^{2}}{L}\left\{2 m\left[E+\frac{G M m}{r}\right]-\frac{L^{2}}{r^{2}}\right\}_{(23.16)}^{\frac{1}{2}}
or, using u=1//ru=1 / r, (u^('))^(2)+u^(2)-(2GMm^(2))/(L^(2))u-(2Em)/(L^(2))=0\left(u^{\prime}\right)^{2}+u^{2}-\frac{2 G M m^{2}}{L^{2}} u-\frac{2 E m}{L^{2}}=0.
Recall that we defined u_(0)=GMm^(2)//L^(2)u_{0}=G M m^{2} / L^{2} and u_(0)(1-epsilon^(2))=-2Em//L^(2)u_{0}\left(1-\epsilon^{2}\right)=-2 E m / L^{2} and wrote
These expressions are consistent with the intuitions we built up about which orbits are possible. In order to compare more closely to the Newtonian version, we write epsilonu_(0)^(2)=(M^(2))/( tilde(L)^(2))( tilde(E)^(2)- tilde(E)_(0)^(2))\epsilon u_{0}^{2}=\frac{M^{2}}{\tilde{L}^{2}}\left(\tilde{E}^{2}-\tilde{E}_{0}^{2}\right) and we have
The terms on the right are relativistic corrections to the Newtonian expression on the left (see Sidenote 8). The Newtonian solution is u=u_(0)(1+epsilon cos phi)u=u_{0}(1+\epsilon \cos \phi). The first-order correction term is 6u_(0)(u-u_(0))^(2)6 u_{0}\left(u-u_{0}\right)^{2}, which is of order O(u_(0)^(3)epsilon^(2))=O(M^(3)epsilon^(2)//r_(0)^(3))O\left(u_{0}^{3} \epsilon^{2}\right)=O\left(M^{3} \epsilon^{2} / r_{0}^{3}\right). In contrast, the second-order term is 2(u-u_(0))^(3)2\left(u-u_{0}\right)^{3}, which is of order O(u_(0)^(3)epsilon^(3))=O(M^(3)epsilon^(3)//r_(0)^(3))O\left(u_{0}^{3} \epsilon^{3}\right)=O\left(M^{3} \epsilon^{3} / r_{0}^{3}\right). For small eccentricities, the second-order correction can be ignored.
From the last example we conclude that the orbits are almost Newtonian ellipses with u=u_(0)(1+epsilon cos phi)u=u_{0}(1+\epsilon \cos \phi), but are slightly perturbed by the corrections terms. Considering only the larger, first-order corrections, we must solve
To solve the equation, define psi=(1-6u_(0))^((1)/(2))phi\psi=\left(1-6 u_{0}\right)^{\frac{1}{2}} \phi and mu=(u-u_(0))\mu=\left(u-u_{0}\right). In terms of these variables, we have an equation of motion
The solution to this equation of motion is periodic in psi\psi. This means that for each orbit Delta psi=2pi\Delta \psi=2 \pi. However, for the angle phi\phi in which we are interested, Delta phi!=2pi\Delta \phi \neq 2 \pi. Instead, from the definition of psi\psi, we can infer that an orbit corresponds to a change in phi\phi of
The geometry for this effect is shown in Fig. 23.4. It implies that the change in the position of the perihelion can be written as Delta phi=2pi+delta phi\Delta \phi=2 \pi+\delta \phi where for each orbit
This provides the basis of our estimate for the precession of the perihelion of mercury.
Example 23.6
The orbit of Mercury has a period of 88 days (~~7.6 xx10^(6)(s))\left(\approx 7.6 \times 10^{6} \mathrm{~s}\right). If we take r_(0)=r_{0}=5.8 xx10^(10)m5.8 \times 10^{10} \mathrm{~m} and M_(o.)=2xx10^(30)kgM_{\odot}=2 \times 10^{30} \mathrm{~kg} then, restoring constants, we compute
This is equivalent to ~~2xx10^(-4)\approx 2 \times 10^{-4} radians per century, or about 41 seconds of arc per century. A more accurate computation (also using general relativity, but with less severe approximations) predicts 43.0 seconds. The observed additional precession is 43.1+-0.543.1 \pm 0.5 seconds. Mercury's orbit actually precesses by around 575 Solar System, and a precession of 532 arcseconds per century could be deduced by including the effects of the other planets. The oblateness of the Sun gives an additional contribution of about 0.03 arcseconds per century. But until the advent of general relativity, the remaining 43 arcseconds per century could not be accounted for, however much people tried to tweak the existing (Newtonian) models. ^(10){ }^{10}
The reason for precession is partly an effect of the curvature of space. This causes the length of the orbiting trajectory to be shorter than our flat-space intuition tells us, so that we can think of there not being enough path length to complete a precession-free Newtonian orbit. However, the larger share of the effect is also due to the curvature effects on the time-coordinate part of the Schwarzschild geometry, so this really is a spacetime effect. ^(11){ }^{11}
Fig. 23.4 The geometry for the precession of the perihelion. ^(10){ }^{10} Others had attempted to resolve the puzzle by tweaking Venus' mass up by 10%10 \%, which was way more than was believable, or by positing a vast swarm lievable, or by positing a vast swarm
of asteroids close to Mercury, which had never been observed. As reported had never been observed. As reported
in Abraham Pais' biography, Einstein in Abraham Pais' biography, Einstein
was strongly affected by the remarkable agreement between the prediction of his theory and the astronomical observations. This instantly resolved a mystery which had stubbornly resisted solution since the 1850s. Einstein wrote to Paul Ehrenfest after his discovery, saying that "For a few days, I was beside myself with joyous excitement," and told another colleague that the discovery had given him heart palpitations. He also said that when he saw the agreement between his calculations and the unexplained observations, he had the feeling that something actually snapped in him. ^(11){ }^{11} One lesson here is that although it is possible to think of curvature in terms of a rubber sheet model, this only tells us about spatial curvature and neglects half of the story. Indeed, the time part is the only part that matters for stationary particles. This point is examined in the exercises.
Chapter summary
Orbits of massive particles in the Schwarzschild geometry can be computed using the conservation of tilde(E)\tilde{E} and tilde(L)\tilde{L} and the identity u^(2)=\boldsymbol{u}^{2}= -1 .
The precession of the perihelion of Mercury is one of the classical tests of general relativity. The theory predicts results in excellent agreement with experiment.
Exercises
(23.1) Consider a planet in a circular orbit, with radius rr, around a spherically symmetric star in the theta=pi//2\theta=\pi / 2 plane of a set of Schwarzschild coordinates.
(a) If the planet has a constant coordinate velocity v=rdphi//dtv=r \mathrm{~d} \phi / \mathrm{d} t, use the line element to find the proper time taken to complete an orbit. Give your answer in terms of vv and rr.
(b) Using the conservation laws in this chapter, show that the proper time to complete a circular orbit is given more generally as Delta tau=\Delta \tau=2pi r((r)/(M)-3)^((1)/(2))2 \pi r\left(\frac{r}{M}-3\right)^{\frac{1}{2}}, and show that this is compatible with the answer in (a).
(23.2) Consider circular motion in a de Sitter geometry with metric
Calculate the proper time taken to complete an orbit, assuming constant velocity v=a(t)Rdphi//dtv=a(t) R \mathrm{~d} \phi / \mathrm{d} t and a factor a(t)=e^(t)a(t)=\mathrm{e}^{t}.
(23.3) For a free particle at a constant radial coordinate in the Schwarzschild geometry, how does the tt variable vary with proper time tau\tau ?
(23.4) A particle moves at a fixed radius in a Schwarzschild geometry at constant angular speed, such that it follows a world line given by
where tau\tau is the proper time and r_(0),Cr_{0}, C and omega\omega are constants.
(a) Use the constraint on the velocity to find an expression for CC.
(b) Compute the components of the acceleration
of the particle.
(c) Under what circumstances does the acceleration vanish?
There are more complex (solved) problems of this sort in Blennow and Ohlsson, from which the preceeding ones have been adapted.
(23.5) The precession of the perihelion results from two curvature effects: g_(tt)g_{t t} affects the rate of clock ticks and g_(rr)g_{r r} leads to a curvature of space. This latter effect can be isolated.
(a) The spatial slice of the Schwarzschild spacetime with t=t= const. and theta=pi//2\theta=\pi / 2 has line element
Hint: This is one of the cases we meet in Appendix DD. The resulting surface is called Flamm's paraboloid [after Ludwig Flamm (1885-1964)]. The particle orbits, not in flat space, but instead on the surface represented by the paraboloid. This resembles, at least approximately, a cone that is tangent to the paraboloid as shown in Fig. 23.5. The distance from the origin in this approximation is RR. For a circular orbit therefore, the apparent flat-space trajectory seems to have length 2pi R2 \pi R, but this is too long when curvature is taken into account. This is because the real radius of the orbit is 2pi r=2pi R cos alpha2 \pi r=2 \pi R \cos \alpha. We must therefore cut out a slice of angle delta\delta to turn the circle of radius RR into the circle of radius rr.
Fig. 23.5 (a) The flat-space radius RR and the real radius R cos alphaR \cos \alpha. (b) A wedge of angular size delta\delta must be cut out of the large circle so that its circumference matches the smaller one.
(b) Using the fact that the tangent to the paraboloid is tan alpha=dz//dr\tan \alpha=\mathrm{d} z / \mathrm{d} r, show that
(c) If the orbit is elliptical, argue that the perihelion must advance by an amount delta\delta on every orbit. (d) How much of the perihelion shift is accounted for by the spatial curvature?
See the book by Moore for more details of this approach.
(23.6) Consider the flat-space (but curved spacetime) metric
This can be used to calculate the contribution to the perihelion shift from the time component of the metric.
(a) Using the usual velocity identity, find an expression for dr//dphi\mathrm{d} r / \mathrm{d} \phi for motion in the equatorial plane in this spacetime.
(b) Show that this leads to an equation of motion
and use this to derive an approximate expression for u_(c)=1//r_(c)u_{\mathrm{c}}=1 / r_{\mathrm{c}} in terms of the constants in the problem.
(d) By expanding in terms of a perturbation to the circular orbit u(phi)=u_(c)+u_(c)w(phi)u(\phi)=u_{c}+u_{c} w(\phi), find an equation for the perturbation w(phi)w(\phi).
(e) Hence, show that the perihelion shift for this metric accounts for 2//32 / 3 of the total perihelion shift for the Schwarzschild metric.
Hint: You should obtain a harmonic oscillator equation for w(phi)w(\phi) with characteristic frequency omega_(0)\omega_{0}. Argue that the distance of closest approach is found when omega_(0)phi=2pi n\omega_{0} \phi=2 \pi n, where nn is an integer, and use this to determine the shift Delta phi\Delta \phi between successive closest approaches. This method can be used to compute the perihelion shift for the Schwarzschild metric and is discussed in many books.
(23.7) Write a simple program to compute trajectories using eqn 23.15 for the case tilde(L)//M=4.3\tilde{L} / M=4.3.
Hint: Consider how to choose the sign in front of the equation, as this changes depending on whether the orbiting particle is heading towards or away from the point of closest approach.
^(1){ }^{1} In cylindrical polar coordinates, straight lines have the equation u=(1)/(r)=-(m cos theta-sin theta)/(c)u=\frac{1}{r}=-\frac{m \cos \theta-\sin \theta}{c},
where mm is the gradient and cc the intercept on the yy-axis. ^(2){ }^{2} Johann Georg von Soldner (17761833). Other neglected predictions of gravitational effects on light rays were made by Henry Cavendish (1731-1810), John Michell (1724-1793) and Newton himself. Michell in particular has been described as one of the greatest unsung scientific heroes of all time, and was also the first person to suggest the existence of black holes, to explain the existence of double stars in terms of grav itation, ad to apply statistics to cos mology. ^(3){ }^{3} Since in the weak-field limit we write sqrt(-g_(00))~~1+2Phi\sqrt{-g_{00}} \approx 1+2 \Phi, we see that this coordinate speed c^(')c^{\prime} varies with the gravitational potential Phi\Phi.
Music is the arithmetic of sounds as optics is the geometry of light.
Claude Debussy (1862-1918)
In Newtonian gravitation, as we have described it, the photon does not interact with the gravitational field and so the paths of all photons in free space are straight lines. ^(1){ }^{1} There had, however, been predictions of the bending of starlight around massive objects, notably by Johann Soldner ^(2){ }^{2} who used Newton's corpuscular theory of light to predict that starlight passing close to the Sun would cause an apparent shift in the position of stars of around 0.8 arc seconds. In this chapter, we see what general relativity has to say about the gravitational interaction of light and matter.
Example 24.1
So why should light rays be affected by gravity? To get an idea, we consider a static, diagonal metric. For the world line of a light ray, we have 0=c^(2)g_(00)dt^(2)+g_(ij)dx^(i)dx^(j)0=c^{2} g_{00} \mathrm{~d} t^{2}+g_{i j} \mathrm{~d} x^{i} \mathrm{~d} x^{j},
That is to say that the speed of light c^(')c^{\prime} determined using the coordinate time varies. ^(3){ }^{3} Although the world line of a light ray is a null geodesic in spacetime, it is not necessarily a geodesic in space. In fact, sqrt(-g_(00))\sqrt{-g_{00}} looks a lot like a refractive index and it can be shown that the spatial part of the trajectory obeys a version of Fermat's principle of least time that says
This provides at least some justification for the bending of light in a static gravitational field.
24.1 Photon trajectories
In general relativity, the curvature of spacetime due to the presence of massive objects causes a photon trajectory to be deflected compared to its path in flat spacetime. One immediate complication with understanding the motion of photons is the impossibility of defining a proper
time tau\tau. Instead we must make use of an affine parameter lambda\lambda that marks off regular intervals along the world line of the photon. Key to computing the influence of gravitation on light is the observation that photons travel along null geodesics. These are paths whose velocity tangent vector u\boldsymbol{u} has the property
where lambda\lambda is an affine parameter. We can write the null condition for photons in the Schwarzschild geometry in order to derive an equation of motion. Written out in terms of conserved quantities ^(4) tilde(E){ }^{4} \tilde{E} and tilde(L)\tilde{L}, the null condition becomes
Multiplying by (1-2M//r)(1-2 M / r), dividing by tilde(L)^(2)\tilde{L}^{2} and following the algebra through, we end up with an effective energy equation for photons, just as we had for massive particles. We cast this in the form E=T+W_("eff ")\mathcal{E}=\mathcal{T}+W_{\text {eff }}, where T\mathcal{T} is the kinetic-energy-like contribution.
The effective energy equation for photons is
where b(-=E^(-1//2))= tilde(L)// tilde(E)b\left(\equiv \mathcal{E}^{-1 / 2}\right)=\tilde{L} / \tilde{E} and there is an effective potential W_("eff ")W_{\text {eff }} for photons of
So the quantity 1//b^(2)1 / b^{2} plays the role analogous to total effective energy E\mathcal{E}, while the role of kinetic energy T\mathcal{T} is taken by (dr//dlambda)^(2)// tilde(L)^(2)(\mathrm{d} r / \mathrm{d} \lambda)^{2} / \tilde{L}^{2}.
Example 24.2
In order to interpret the physical meaning of bb, consider the geometry in Fig. 24.1. A photon is initially moving parallel to the xx-axis, a perpendicular distance dd away from it, such that y=dy=d at infinity. This defines the impact parameter dd. Far from any source of curvature the photon moves in a straight line, so for r≫2Mr \gg 2 M we have
We conclude that b=db=d. That is, bb is the impact parameter for a light ray that reaches infinity.
The definition of bb makes sense physically in that the impact parameter is a measure of the angular momentum: it evaluates how far from the ^(4){ }^{4} The conserved quantities followed from geometric considerations in the last chapter and so apply for photons as well as for massive particles, although we cannot parametrize the world lines of photons with the proper time.
Fig. 24.1 The geometry defining the impact parameter.
Fig. 24.2 The relativistic effective potential for photons. ^(5){ }^{5} This step involves a little rearrangement of the indices
origin a trajectory is, but decreases with increasing energy, since there is less scattering of an energetic particle by the gravitational potential compared to one with less energy.
The most important quantity in understanding the trajectories of light rays is the effective potential W_("eff ")W_{\text {eff }} which is shown in Fig. 24.2. Unlike the analogous case for massive particles, the effective potential is independent of tilde(L)\tilde{L} for photons. Like the relativistic V_("eff ")V_{\text {eff }}, the potential W_("eff ")W_{\text {eff }} has a maximum; this occurs at r_(0)=3Mr_{0}=3 M where W_("eff ")(r_(0))=(1)/((27M^(2)))W_{\text {eff }}\left(r_{0}\right)=\frac{1}{\left(27 M^{2}\right)}. When an incoming photon has a large impact parameter, such that 1//b^(2) < W(r_(0))=1//27M^(2)1 / b^{2}<W\left(r_{0}\right)=1 / 27 M^{2}, then the photon will always be deflected by the potential. However, if an incoming photon has a small enough impact parameter such that 1//b^(2) > W(r_(0))1 / b^{2}>W\left(r_{0}\right) then the photon will spiral in towards r=0r=0 where it is destroyed. Also unlike both the relativistic and Newtonian effective potentials for massive particles, W_("eff ")W_{\text {eff }} has no local minimum, so we should not expect any stable circular orbits for photons. However, at the maximum at r_(0)r_{0} circular orbits of light are possible, with impact parameter b^(2)=27M^(2)b^{2}=27 M^{2}, in a region of space called the photon sphere. Since this solution lies at the maximum of W_("eff ")W_{\text {eff }} these trajectories are unstable.
Using the same techniques as we employed for massive particles, we can come up with an equation of motion for the photons, again using the variable u=M//ru=M / r.
Example 24.3
Let's examine light rays in the theta=pi//2\theta=\pi / 2 plane so we have that u^(theta)=0u^{\theta}=0. The equation providing the trajectory (the change in radius with phi\phi ) is given by
The first term can be rewritten ^(5){ }^{5} using (u_(t)//u_(phi))^(2)=1//b^(2)\left(u_{t} / u_{\phi}\right)^{2}=1 / b^{2}. On inserting the metric components, we have
The Newtonian analogue of this equation can be extracted from eqn 20.33 by setting m=0m=0, yielding u^('')+u=0u^{\prime \prime}+u=0. The right-hand side of eqn 24.16 therefore provides the relativistic correction.
The equation of motion allows us to evaluate how light rays are deflected by gravitation. In fact, this effect is the second of our classical solarsystem tests of general relativity. The geometry for light deflection is shown in Fig. 24.3.
Example 24.4
Consider the equation of motion. In the limit M//b≪1M / b \ll 1, we expect the straight line solution
which solves the Newtonian equation u_(0)^('')+u_(0)=0u_{0}^{\prime \prime}+u_{0}=0. We can expand the equation of motion in terms of perturbations to this straight line as u(phi)~~u_(0)(phi)+u_(1)(phi)u(\phi) \approx u_{0}(\phi)+u_{1}(\phi). Doing this in eqn 24.16, we find
where we have retained only the largest term on the right (i.e. u_(0)u_{0} ) in the first of these expressions. The solution to this equation is given by
Angles that approximately satisfy this condition are found at phi~~\phi \approx-2(M//b)-2(M / b) and phi~~pi+2(M//b)\phi \approx \pi+2(M / b). Our conclusion is that the total deflection angle Delta phi\Delta \phi for light is therefore 4M//b4 M / b.
Example 24.5
For light that just grazes the Sun, ^(6){ }^{6} we have b=6.9 xx10^(8)mb=6.9 \times 10^{8} \mathrm{~m} and we compute a deflection (with constants restored) of
which is equivalent to a prediction of 1.8 arc seconds (around twice Soldner's prediction based on Newtonian corpuscular theory). This was the prediction of general relativity that was tested by Arthur Eddington's 1919 observation of the shift in stellar positions during a solar eclipse, when starlight had passed close to the Sun would become temporarily visible. ^(7){ }^{7} The expedition gave measurements that agreed with the prediction within estimated experimental uncertainties. This made Einstein an internationally celebrated figure. Some later analyses of the Eddington results have questioned the systematic errors involved in the measurements. Later high-precision measurements using radio sources have confirmed that the measured shifts are in excellent agreement with the predictions of general relativity (and the picture shown in Fig. 24.4 is a common way of illustrating the effect).
Fig. 24.3 The geometry for the deflection of a photon. ^(6){ }^{6} The solar radius is usually taken to be ~~696,000km\approx 696,000 \mathrm{~km}. ^(7){ }^{7} For an account of the history, see J. Earman and C. Glymour, Historical Studies in the Physical Sciences 11, 175 (1980).
Fig. 24.4 An illustration of how curved space bends starlight around the Sun.
Fig. 24.5 (a) Light bent by a gravitating mass is not focussed to a point. (b) An 'Einstein ring' is formed by the deflection of light originating from a source ss around a mass at ℓ\ell. The trajectory of the light is such that an observer at OO will trace the rays back via straight lines to form the circular image in the same plane as ss. ^(8){ }^{8} The consequences of this are examined in the exercises.
The deflection of light rays due to gravity allows us to treat gravitating masses a little like lenses. However, as discussed in the next example, the analogy is not an exact one.
Example 24.6
Consider the setup in Fig. 24.5(a) where light coming from infinity is bent by a gravitating mass at ℓ\ell to reach the observer. Working in the small-angle approximation, light is detected a distance DD from ℓ\ell where D=b//thetaD=b / \theta. The deflection angle theta\theta for a gravitating mass MM is 4M//b4 M / b and so we obtain
{:(24.24)D=b^(2)//4M:}\begin{equation*}
D=b^{2} / 4 M \tag{24.24}
\end{equation*}
This shows that the light rays are focussed down to different points depending on how far they are from the axis. They are not, therefore, focussed down to a point. This is examined further in the exercises.
In the case that the source, lens, and observer are all in a line, the image takes the form of a ring, ^(8){ }^{8} known as the Einstein ring, as demonstrated in Fig. 24.5(b).
24.2 Looking around
The passage of light rays in a gravitational field determines what observers can see. To investigate this we must consider the momentum of photons. Recall that owing to photons being massless, the components of the photon momentum vector in flat spacetime are related to the velocity components of the photon via p^(mu)=Eu^(mu)p^{\mu}=E u^{\mu}, where EE is the photon energy. Our first task is to update this for the curved spacetime we experience in the Schwarzschild geometry.
Example 24.7
A momentum vector p\boldsymbol{p} for a particle of mass mm has components in the coordinate frame of
This is well defined in the limit m rarr0m \rightarrow 0 and so is suitable as a definition of the momentum components for the photon. The price we have paid is the inclusion of the coordinate tt. In a coordinate system (t,r,theta,phi)(t, r, \theta, \phi), the momentum components become
We also have b=r^(2)(1-2M//r)^(-1)dphi//dtb=r^{2}(1-2 M / r)^{-1} \mathrm{~d} \phi / \mathrm{d} t and, from the null condition for photons, we can write
{:[(24.30)0=-(1-(2M)/(r))+(1-(2M)/(r))^(-1)(((d)r)/((d)t))^(2)+r^(2)(((d)phi)/((d)t))^(2)],[(24.31)" or "],[qquad1=(1-(2M)/(r))^(-2)(((d)r)/((d)t))^(2)+(1-(2M)/(r))(b^(2))/(r^(2))],[" which gives us an expression for "dr//dt". Collecting these facts, the final result for the "]:}\begin{align*}
& 0=-\left(1-\frac{2 M}{r}\right)+\left(1-\frac{2 M}{r}\right)^{-1}\left(\frac{\mathrm{~d} r}{\mathrm{~d} t}\right)^{2}+r^{2}\left(\frac{\mathrm{~d} \phi}{\mathrm{~d} t}\right)^{2} \tag{24.30}\\
& \text { or } \tag{24.31}\\
& \qquad 1=\left(1-\frac{2 M}{r}\right)^{-2}\left(\frac{\mathrm{~d} r}{\mathrm{~d} t}\right)^{2}+\left(1-\frac{2 M}{r}\right) \frac{b^{2}}{r^{2}} \\
& \text { which gives us an expression for } \mathrm{d} r / \mathrm{d} t \text {. Collecting these facts, the final result for the }
\end{align*} momentum components of light is
The geometry for photons emitted from a point at some radius rr is shown in Fig 24.6. The most striking thing here is that photons emitted beyond a particular angle psi\psi will have (1//b^(2)) > W(r_(0))\left(1 / b^{2}\right)>W\left(r_{0}\right) (see Fig. 24.7) and will therefore spiral into the origin, where they are destroyed. The function sin psi\sin \psi equals the sideways component of the coordinate velocity of the photon. Referring to the figure, we see that, in the observer's local frame, this can be written as
where, in the second step, we've used p^( hat(mu))=Eu^( hat(mu))p^{\hat{\mu}}=E u^{\hat{\mu}} in the local frame. In order to use this, we need the conventional vielbein components for an orthonormal frame ^(9){ }^{9} to shift components via
The geometry is such that if the angle psi\psi is greater than a critical angle psi_(c)\psi_{\mathrm{c}} then 1//b^(2) > W(r_(0))1 / b^{2}>W\left(r_{0}\right) and the photon will spiral into the origin. This critical angle occurs when 1//b^(2)=W(r_(0))1 / b^{2}=W\left(r_{0}\right), or b=sqrt27Mb=\sqrt{27} M. [When the equality holds we expect the photons to (unstably) orbit the central mass.] Substituting, we conclude that those photons with angles greater than sin psi_(c)=(sqrt27M//r)(1-2M//r)^((1)/(2))\sin \psi_{\mathrm{c}}=(\sqrt{27} M / r)(1-2 M / r)^{\frac{1}{2}} are condemned to fall into the central mass, where they are destroyed. Photons with angles psi < psi_("c ")\psi<\psi_{\text {c }}
Fig. 24.6 The geometry for photons emitted from a point at radius rr.
Fig. 24.7 A photon with 1//b^(2) >1 / b^{2}>W_("eff ")(r_(0))W_{\text {eff }}\left(r_{0}\right) will spiral into the origin of the coordinate system. ^(9){ }^{9} The vielbein components for theta=pi//2\theta=\pi / 2 are (e_(t))^( hat(t))=(1-(2M)/(r))^((1)/(2))\left(\boldsymbol{e}_{t}\right)^{\hat{t}}=\left(1-\frac{2 M}{r}\right)^{\frac{1}{2}}, (e_(r))^( hat(r))=(1-(2M)/(r))^(-(1)/(2))\left(\boldsymbol{e}_{r}\right)^{\hat{r}}=\left(1-\frac{2 M}{r}\right)^{-\frac{1}{2}}, (e_(theta))^( hat(theta))=r\left(e_{\theta}\right)^{\hat{\theta}}=r, (e_(phi))^(phi)=r\left(\boldsymbol{e}_{\phi}\right)^{\boldsymbol{\phi}}=r.
escape, but as the radius rr decreases, more and more of the photons are captured since the critical angle becomes smaller. The critical angle vanishes for r=2Mr=2 M and all photons are captured.
The Schwarzschild geometry is invariant with respect to time reversal. That is, if we play a film of the trajectories in reverse, the geometry is unchanged and so this same physics largely applies to the photons received by an observer at position rr. The exception is the behaviour of those photons that were destroyed by spiralling into the origin: when we play the film backwards, these photons are not emitted and that part of space at angles psi > psi_(c)\psi>\psi_{c} is black. We will see the consequence of this in the next chapter, whose subject is black holes.
Chapter summary
Photons are deflected by gravitating objects in general relativity.
An effective-energy equation allows deflection to be calculated.
The deflection of light by gravitating stars is one of the classical tests of general relativity. The theory predicts results in excellent agreement with experiment.
It is useful at this stage to summarize the equations for motion in the Schwarzschild geometry for massive particles and for photons, where we assume the motion takes place in the equatorial plane.
(24.1) Confirm eqns 24.6 and 24.7 .
(24.2) Confirm that eqn 24.21 solves eqn 24.19.
Fig. 24.8 Ray diagram for Exercise 24.3, showing a source ss, a lensing mass ℓ\ell and an observer OO.
(24.3) Consider the optical diagram in Fig. 24.8.
(a) Using the small-angle approximation, show that
where we have defined the Einstein angle theta_(E)^(2)=\theta_{\mathrm{E}}^{2}=4MD_(ℓs)//D_(ℓ)D_(s)4 M D_{\ell \mathrm{s}} / D_{\ell} D_{\mathrm{s}}.
(b) Find an expression of the angular size of the Einstein ring for the case that the source is at infinity and the source, lensing mass and observer are collinear.
(24.4) A light ray travels along a radial line in a spacetime described by the Robertson-Walker line element
where dOmega^(2)=dtheta^(2)+sin^(2)thetadphi^(2)\mathrm{d} \Omega^{2}=\mathrm{d} \theta^{2}+\sin ^{2} \theta \mathrm{~d} \phi^{2}.
(a) Show that, when considering trajectories along the ray, the metric can be rewritten as
The knowledge of how the affine parameter varies along a geodesic is often useful. See Exercise 26.4 for an example.
25
25.1 The surface r=2M quad264r=2 M \quad 264 25.2 The tortoise coordinate 265 25.3 Death of an astronaut 266 25.4 Looking around near a black
hole hole
25.5 Gravitational collapse 268 Chapter summary 270 Exercises 270 ^(1){ }^{1} John Michell, and subsequently Pierre-Simon Laplace (1749-1827), had suggested the possibility of a dark star whose gravitation would be such that not even light would have the necessary velocity to escape the star's surface. The modern concept of a black hole was investigated by Robert Oppenheimer (1904-1967) and coworkers in the late 1930s, who proposed that the gravitational collapse of neutron stars could potentially cause them to become black holes (although the term 'black hole' was not used at this point; it was popularized by John Wheeler in the 1960 s). The key paper Wheeler in the 1960 s ). The key paper
on gravitational collapse, co-authored on gravitational collapse, co-authored
by Oppenheimer and Hartland Snyder (1913-1962) was published in 1939 on the eve of World War II, a conflict in which Oppenheimer would, of course, play a significant role. ^(2){ }^{2} The radius r_(S)r_{\mathrm{S}} corresponds to a spherical shell of singularities in (3+1)dimensional spacetime. Since we usually draw space on a two-dimensional page, it is sometimes called a a 'ringlike' singularity. The Schwarzschild singularity is sometimes also called the 'Hadamard catastrophe', as this is what Einstein jokingly called it, referring to the Jacques Hadamard's (1865-1963) the Jacques Hadamard's (1865-1963)
suggestion that it could be regarded resuggestion
alistically. ^(3){ }^{3} That is, frequency goes to zero since an oscillation takes infinite time.
Black holes
When black clouds envelop stars which shone bright,
They can no longer pour forth their light.
Boethius (c.480-c.524/6) The Consolation of Philosophy
A black hole ^(1){ }^{1} is a part of spacetime where gravitational effects are so severe that light cannot escape from the inside of the region. It is likely that a black hole is the state of matter that, if massive enough, a star can adopt at the end of its life. Spacetime with a non-rotating black hole at its centre is spherically symmetric and must therefore, by Birkhoff's theorem, be described by the Schwarzschild metric.
The Schwarzschild geometry for the spacetime outside a gravitating object with mass MM is represented by the line element
It is notable that the Schwarzschild line element looks to be very badly behaved at two coordinate values: r=0r=0 and also at r=r_(S)=2Mr=r_{\mathrm{S}}=2 M, where r_(S)r_{\mathrm{S}} is known as the Schwarzschild radius. ^(2){ }^{2} On the three-dimensional spherical surface defined by r_(S)r_{\mathrm{S}}, trouble comes from two places: (i) the vanishing of the first term tells us that clocks at the Schwarzschild radius measure no time, which causes light signals to be infinitely redshifted; ^(3){ }^{3} (ii) the divergence of the second term means that del s//del r\partial s / \partial r is infinite: a small change in rr leads to a divergent change in the invariant interval ds\mathrm{d} s. When Schwarzschild's solution was first discussed, it was expected that since the singularity at r=0r=0 was hidden deep inside objects like stars, its effects should be unobservable. The singularity at r_(S)r_{\mathrm{S}} originally had a similar status: if we restore units and plug numbers in, we find
{:(25.2)r_(S)=(2GM)/(c^(2))~~(3M)/(M_(o.))km:}\begin{equation*}
r_{\mathrm{S}}=\frac{2 G M}{c^{2}} \approx \frac{3 M}{M_{\odot}} \mathrm{km} \tag{25.2}
\end{equation*}
where M_(o.)~~2xx10^(30)kgM_{\odot} \approx 2 \times 10^{30} \mathrm{~kg} is the solar mass. For objects with masses of order a solar mass, it appeared that this badly behaved point was buried well within the innards of the star. However, the discovery of incredibly massive, dense objects with very strong gravitational fields has forced relativists to confront the details of the singular coordinate r_(S)r_{\mathrm{S}} head on. Specifically, the mass distribution for a black hole is actually confined to an infinitely dense point at the origin, with the result that the region 0 < r < 2M0<r<2 M lies outside of the mass that forms the hole.
It is the behaviour of massive particles and of light close to the radius r_(S)=2Mr_{\mathrm{S}}=2 M that leads to many of the most notable features of a black hole. We shall see that although the metric at r_(S)=2Mr_{\mathrm{S}}=2 M has several interesting features, it does not lead to any bad behaviour in the local physical fields. The curvature of spacetime, for example, varies smoothly over this region. An astronaut travelling towards the origin would not notice anything strange in their trajectory as they passed this point, with their watch continuing to tick regularly as they cross the surface r=r_(S)r=r_{\mathrm{S}} in a finite interval of proper time. (Indeed, we saw in Chapter 22 that any point in the geometry can be reached, via a radial plunge, in a finite interval of proper time.) In fact, the threat of an infinity at r_(S)r_{\mathrm{S}} is actually the consequence of our choice of coordinates in the Schwarzschild geometry and much of our task of the next chapter will be to identify a set of coordinates that elucidates the behaviour of light and particles near black holes. In this chapter, our task is to examine the behaviour of geodesics that pass close to r=2Mr=2 M using the Schwarzschild geometry. ^(4){ }^{4}
Example 25.1
What is the evidence for the existence of black holes? As they cannot be directly seen (since, of course, light cannot escape their vicinity), the evidence for them is necessarily somewhat indirect. The observational evidence falls into four main categories.
Stars close to the centre of the Milky Way have proper motions that indicate that they are whirling around a very compact radio source, believed to be a supermassive black hole of mass around 4xx10^(6)M_(o.)4 \times 10^{6} M_{\odot}, named Sagittarius A^(**)A^{*}. The accretion disc around this was imaged by the Event Horizon Telescope in 2022 (see Fig. 25.1).
Binary star systems that are strong sources of XX-rays. The mass of an unseen partner in a binary star system can be determined by its influence on the visible partner and can be inferred to be too massive to be a white dwarf or a neutron star. The X-ray emission occurs owing to the mass at the centre of the unseen part of the binary accreting an energetic disc of massive material from its partner. This matter orbits around the mass (at a distance larger than the event horizon, see below). Owing to the large velocities of the matter in the accretion disc, (which are caused by the large amount of angular momentum involved in the formation of the disc), the material in the disc has a temperature high enough to radiate strongly in the X-ray region of the electromagnetic spectrum.
The most famous example is Cygnus-X1, an X-ray source found in the Cygnus constellation. The object is partnered with the blue supergiant HDE 226868 in a binary system. The X-ray emission is consistent with a source with diameter smaller than several hundred kilometres. The mass at the centre of Cygnus-X1 is estimated to be ~~15M_(o.)\approx 15 M_{\odot}, which is far too large to be a neutron star.
Anomalous behaviour close to the centre of a galaxy, that can be explained by an enormous, massive, but unseen, object. Such objects can also be sources of large amounts of energy. Quasars ^(5){ }^{5} are very luminous (and therefore hugely energetic) objects that lie at the centre of galaxies. They are thought to comprise a supermassive black hole (i.e. black holes of millions to billions of M_(o.)M_{\odot} ) surrounded by an accretion disc. The most distant quasar known is ULAS J1342+0928, which is expected to comprise the oldest known supermassive black hole, with a mass estimated at 8xx8 \times10^(8)M_(o.)10^{8} M_{\odot}.
Gravitational wave observations that can be explained by the collision of black holes. We will return to these when we discuss gravitational waves in Chapter 46. ^(4){ }^{4} It's perhaps worth stressing that the picture of a black hole as an entity relentlessly sucking in all of the matter in the Universe is not an accurate one. Its exterior geometry is no different to that of any other star. As long as an observer has access to some means of propulsion, they can always escape a black hole (although, as we shall see, this is as long as they never get closer than r=r_(S)r=r_{\mathrm{S}} ).
Fig. 25.1 An image of the supermassive black hole, Sagittarius A*, at the centre of the Milky Way, captured by an array of eight radio observatories distributed around Earth, networked together as the Event Horizon Telescope (EHT). The image shows a dark central region (called a shadow) surrounded by a bright ring-like structure. A mass of 4xx10^(6)M_(o.)4 \times 10^{6} M_{\odot} is shoehorned into a region of space of diameter of only 0.3 AU. (Courtesy ESO.) ^(5){ }^{5} This curious word is a contraction of quasi-stellar radio sources. As the name suggests, they were originally thought to be stars. In spite of this, their large energies, large redshifts and the jets that they can emit are good evidence that they cannot be stars. ^(6){ }^{6} As we saw in Chapter 22, coordinate time and the proper time of the observer travelling towards the surface are very different. tt ii
Fig. 25.2 Starting at large rr, light cones eclipse as rr decreases on the approach to a black hole, before turning over for r < r_(S)r<r_{\mathrm{S}}. ^(7){ }^{7} This is a vivid manifestation of the lack of any metric significance of coordinates in general relativity: a variable called tt has no more claim to tell the time than one called uu or chi\chi. ^(8){ }^{8} This section should therefore really have been called 'Inside and after the surface r=2Mr=2 M,
25.1 The surface r=2Mr=2 M
Let's consider the regions close to the spherical surface defined by r_(S)=r_{\mathrm{S}}=2M2 M. First, we look at the geometry outside this surface, where r > 2Mr>2 M. Light cones in the Schwarzschild geometry, like all light cones, have ds^(2)=0\mathrm{d} s^{2}=0, so we have from eqn 25.1 that, for constant theta\theta and phi\phi, they are described by
The light cones have slope +-1\pm 1 for large rr where spacetime becomes (asymptotically) flat. As we approach the surface r_(S)r_{\mathrm{S}} from r > 2Mr>2 M (denoted r rarr2M^(+)r \rightarrow 2 M^{+}below) the slope of the light cone approaches +-oo\pm \infty (meaning that the null surfaces of the cone point vertically in a standard rr vs. tt plot), closing up (or eclipsing) as shown in Fig. 25.2. As always, massive particles cannot travel faster than light, so are confined to the timelike part of the cone. The consequence of the narrowing of the light cones as r rarr2M^(+)r \rightarrow 2 M^{+}is that the trajectories of particles become more vertical in the rr - tt plane, which is to say that rr cannot change as much for a given time interval. Therefore, as the particle approaches r_(S)=2Mr_{\mathrm{S}}=2 M it seems to take longer and longer in coordinate time tt to make a change in rr. In fact, this is the effect we saw in Chapter 22 where it takes infinite amount of coordinate time to reach the surface at r_(S)r_{\mathrm{S}} during a radial plunge. ^(6){ }^{6}
Let's now examine the geometry of spacetime just inside the surface at r_(S)r_{\mathrm{S}}. Introducing the coordinate epsi\varepsilon via r=2M-epsir=2 M-\varepsilon, the line element can be written as
We see that if we fix t,thetat, \theta and phi\phi at constant values, the interval is ds^(2)=\mathrm{d} s^{2}=-(2M-epsi)depsi^(2)//epsi-(2 M-\varepsilon) \mathrm{d} \varepsilon^{2} / \varepsilon. Remarkably, despite our having fixed things so that dt=0\mathrm{d} t=0, this interval has ds^(2) < 0\mathrm{d} s^{2}<0, which is to say that it is timelike! This implies that epsi\varepsilon (the 'radial' coordinate inside the surface) is timelike, rather than spacelike as we expect for spatial coordinates. A timelike epsi\varepsilon is doomed to constantly increase (like time in flat spacetime), causing the radial coordinate rr to decrease until eventually we meet the singularity at r=0r=0. By the same token, the 'time' coordinate tt in eqn 25.4 has become spacelike ^(7){ }^{7} inside the surface r_(S)=2Mr_{\mathrm{S}}=2 M.
A physical consequence of this topsy-turvy state of affairs can be seen by considering an astronaut at radius r < 2Mr<2 M who sends out a photon. As shown in Fig. 25.2, the light cones inside r=2Mr=2 M tip over, spitting particles towards the singularity at r=0r=0. That is to say that the photon must go forward in time, which means rr decreases and the photon has no option but to fall towards the origin. This means that for any observer within the event horizon, the future points towards r=0r=0 or, to sloganize: the future is inwards. ^(8){ }^{8} Photons inside the surface r=2Mr=2 M are therefore trapped and, as a consequence, so are all massive particles. With the impossibility of photons escaping the r_(S)r_{\mathrm{S}} surface, it is impossible
for anything inside r_(S)r_{\mathrm{S}} to be seen. In the same way that we cannot see beyond the Earth's horizon, we call the surface at r_(S)r_{\mathrm{S}} the event horizon of the black hole.
Taking the above considerations into account, we can plot the light cones in Schwarzschild coordinates, which are shown in Fig. 25.3. The light cones have the alarming feature that null trajectories appear singular at r_(S)r_{\mathrm{S}} with incoming trajectories shooting off to infinite tt before returning. It will turn out that this is not, in fact, real physical behaviour and in the next chapter we shall use an alternative coordinate description to understand the illusory nature of this feature.
25.2 The tortoise coordinate
The investigation of black holes in general relativity is, in large part, an exercise in finding the right coordinates with which to describe them. In evaluating the coordinate time tt measured by some observer, we will often have cause to integrate equations like eqn 25.3 . Integrals of the form
lead to logarithmic contributions to the coordinate-time interval. To understand these logarithms, an interesting coordinate that we can employ is the tortoise coordinate r^(**)r^{*}, defined by
This coordinate is named in honour of the story of Achilles ^(9){ }^{9} who races the tortoise and, despite moving at a high speed, apparently can never overtake it. This is because when Achilles catches up with the tortoise, the tortoise has moved to a new location. Achilles catches up again, but again, by the time he does, the tortoise has moved a bit further along the race track.
In our coordinates, we imagine a rather contrived race between Achilles, whose separation from the tortoise is given by d=r-r_(S)d=r-r_{\mathrm{S}}, so that when r=r_(S)=2Mr=r_{\mathrm{S}}=2 M, Achilles expects to pass the tortoise. We track progress in the race using the coordinate r^(**)r^{*} which one can think of as a curious sort of race timer, that starts at positive values of r^(**)r^{*} and monotonically decreases, with the possibility of taking all values -oo <= r^(**) <= oo^(10)-\infty \leq r^{*} \leq \infty^{10} We start with r^(**)r^{*} taking a large, positive value, corresponding to the athletes being well separated (i.e. large d=r-r_(S)d=r-r_{\mathrm{S}} ). As we time-evolve the race, we allow the timer r^(**)r^{*} to decrease, and Achilles closes in on the tortoise, with rr approaching r_(S)r_{\mathrm{S}}. As the race coordinate r^(**)r^{*} becomes large and negative and rr gets close to r_(S)r_{\mathrm{S}}, we see from eqn 25.6 that rr changes more and more slowly with the race time r^(**)r^{*}, since the race velocity dr//dr^(**)rarr0\mathrm{d} r / \mathrm{d} r^{*} \rightarrow 0. We continue to time-evolve the race, making the r^(**)r^{*} coordinate more and more negative. However, Achilles never reaches the tortoise.
Fig. 25.3 Light cones in the Schwarzschild geometry.
Fig. 25.4 A plot of eqn 25.7. ^(9){ }^{9} Achilles was a hero of the Trojan war and has a starring role in Homer's Iliad. Achilles is played by Brad Pitt in the film Troy, and even appears in the title of the Led Zeppelin track 'Achilles Last Stand'. The philosopher Zeno of Elea (Fifth century bc) picked Achilles for his paradox because, with regards to land speed, the slowly moving tortoise would be expected to be no match for the supremely athletic Achilles. ^(10){ }^{10} Since coordinates have no intrinsic metric significance, we don't worry about exactly what this means in terms of the workings of the clock.
Fig. 25.5 Motion of an astronaut falling into a black hole as a function of proper time.
Fig. 25.6 Motion of an astronaut falling into a black hole as a function of Schwarzschild coordinate time. ^(11){ }^{11} The equation of motion for the separation of two point is given, in the orthonormal frame, by
Notice that the curvature has no notable behaviour at r=2Mr=2 M.
The tortoise coordinate is useful as it can be used to prevent the eclipse of the light cones that we saw occurring in the Schwarzschild coordinates as r rarr2M^(+)r \rightarrow 2 M^{+}owing to the property that dt//dr=+-(1-2M//r)^(-1)rarr oo\mathrm{d} t / \mathrm{d} r= \pm(1-2 M / r)^{-1} \rightarrow \infty. The solution is to note that the light cones can be described by setting
with rr a function of r^(**)r^{*}. This solves the problem of the light cones closing at r=2Mr=2 M and it also prevents the metric having any bad behaviour around r=2Mr=2 M. Nonetheless, it is not an easy set of coordinates to work with, since the point r=2Mr=2 M now occurs at infinite r^(**)r^{*}. We return to the problem of finding better coordinates in the next chapter.
25.3 Death of an astronaut
We observe an astronaut undergoing a radial plunge towards a black hole. She has a torch that emits light pulses. We saw in the last chapter that, from her point of view, the proper time during a radial plunge changes with the rr coordinate via
The proper time for the astronaut to fall through the horizon is finite, and is shown by a solid line in Fig. 25.5. The astronaut notices nothing special about her clock as she passes the point of no return at r_(S)r_{\mathrm{S}}.
However, from our point of view as observers at spatial infinity, our spacetime is flat and our proper time coincides with the coordinate time tt. Using the result from the last chapter, the coordinate time during a radial plunge evolves according to
This quantity diverges as r rarrr_(S)r \rightarrow r_{\mathrm{S}} and so, we see the astronaut taking an infinite amount of time to cross the boundary. This is shown in Fig. 25.6.
Example 25.3
What does the astronaut feel as she falls? ^(11){ }^{11} We evaluated the forces that act on an astronaut in the Schwarzschild geometry in the orthonormal frame in Chapter 11. These are
The astronaut is stretched out like spaghetti in the hat(r)\hat{r} direction, and compressed in the hat(theta)\hat{\theta} and hat(phi)\hat{\phi} directions. Notice that there are no special forces that the astronaut feels as she passes r_(S)r_{\mathrm{S}}. Nevertheless, the astronaut is doomed to be stretched out (and compressed) to death by the forces whose magnitudes all diverge as r rarr0r \rightarrow 0.
The last example suggests that, whatever the status of the bad behaviour at r_(S)r_{\mathrm{S}}, we should take the singular behaviour of the metric at r=0r=0 very seriously. This is indeed the case as the point r=0r=0 does represent a physical singularity. Any body that meets the point r=0r=0 must be destroyed by the infinite forces at this point. Meeting this point is the fate of anything that finds itself within the event horizon of a black hole.
What about the light pulses that the astronaut emits? Light rays travel along null geodesics and so, assuming the pulses are emitted radially, the interval between two events on the world line of the photons emitted by the astronaut are given by
The photons of interest are those travelling radially outwards, so we choose the +sign+\operatorname{sign} (i.e. rr increases as tt increases). The journey time for a photon emitted at (t_(1),r_(1))\left(t_{1}, r_{1}\right) and detected at (t_(2),r_(2))\left(t_{2}, r_{2}\right) is
The coordinate-time interval is therefore corrected from the usual (flat space) value ( r_(2)-r_(1)r_{2}-r_{1} ), with the journey time increased by the logarithmic correction (Fig. 25.7). The logarithm gets larger and larger as the point of emission r_(1)r_{1} gets closer to r_(S)r_{\mathrm{S}}, causing the interval to diverge. As seen by us distant observers, the light pulses from the astronaut become less and less frequent. If we are able to see the astronaut during her descent then she will appear to never quite reach the horizon owing to the coordinate time interval getting larger and larger as we saw above. In addition to the light pulses emitted from the astronaut becoming less frequent, they will also become dimmer as light is severely redshifted by the gravitational effect. ^(12){ }^{12} ^(12){ }^{12} Since this observer is falling, we cannot simply use the result from eqn 22.16 which assumes a stationary observer. In fact, we will examine just how severe the redshift in the next chapter after we have identified a more suitable set of coordinates to describe the situation.
25.4 Looking around near a black hole
What does the astronaut see as she plunges towards the hole? Recall from the previous chapter that there is a critical angle for photons
Fig. 25.7 Equation 25.15, giving an interval Delta t=t_(2)-t_(1)\Delta t=t_{2}-t_{1}, plotted as a function of r_(2)r_{2} for given values of r_(1)r_{1}. ^(13){ }^{13} Interestingly, since when the equality holds we expect the photons to or bit the central mass, photons emitted from an observer at r=3Mr=3 M and psi_(c)\psi_{c} can be observed by that same observer after completing an orbit. This implies that the astronaut can see the back of her head. ^(14){ }^{14} The use of the vielbein components here is equivalent to the usual rule that E=-p*u_("obs ")E=-\boldsymbol{p} \cdot \boldsymbol{u}_{\text {obs }} that we used to work out the gravitational redshift in Chapter 22. Recall also that our conventional vielbein corresponds to a stationary observer's local rest frame. ^(15){ }^{15} That is, p_(t)(r)=(1-2M//r)p^(t)(r)p_{t}(r)=(1-2 M / r) p^{t}(r) is conserved. Therefore, p^(t)(oo)=ℏomega_(oo)=p^{t}(\infty)=\hbar \omega_{\infty}=(1-2M//r)p^(t)(r)(1-2 M / r) p^{t}(r), so p^(t)(r)=ℏomega_(oo)(1-p^{t}(r)=\hbar \omega_{\infty}(1-2M//r)^(-1)2 M / r)^{-1}.
Fig. 25.8 Gravitational collapse with two spatial dimensions suppressed. Each point represents a 2-sphere. ^(16){ }^{16} Subrahmanyan Chandrasekhar (1910-1995) computed that a white dwarf with a mass ≳1.4M_(o.)\gtrsim 1.4 M_{\odot} (the Chandrasekhar limit) would be unstable to further gravitational collapse. This results in a neutron star or, if the star is still more massive, a black hole. The most massive known neutron star is ~~2.4M_(o.)\approx 2.4 M_{\odot}; the least massive black hole is thought to be ~~4M_(o.)\approx 4 M_{\odot}.
emerging from a point at radius rr in the Schwarzschild geometry, given by
At angles greater than psi_(c)\psi_{c}, the photons spiral into the origin and are captured. Notice how the angle decreases to zero for r=2Mr=2 M : we cannot see any light signal from this point as all photons are captured.
The Schwarzschild metric is invariant with respect to time reversal, which means that this argument also applies to photons arriving at the location rr where the astronaut finds herself. This affects what the astronaut observes: photons from r <= 2Mr \leq 2 M cannot reach her since photons do not emerge from a black hole. For photons from elsewhere, psi_(c)\psi_{\mathrm{c}} represents a limit to what the astronaut outside the hole can see, with incoming photons from angles less than psi_(c)\psi_{c} being the only ones that she is able to detect. The rest of her field of vision is black: this is the astronaut 'seeing' the black hole.
Example 25.4
For example, when r=3Mr=3 M we have sin psi_(c)=1\sin \psi_{\mathrm{c}}=1 and psi_(c)=pi//2\psi_{c}=\pi / 2. The black hole then occupies exactly half of the sky. ^(13){ }^{13} As the astronaut gets closer to the black hole, the hole takes up more and more of the sky, until, very close to the horizon, light from the rest of the universe is only experienced through a small cone.
The photons that do reach the astronaut from other stars have their energies increased (or blueshifted) compared to the energies of these photons at infinity. This can be seen by evaluating ^(14){ }^{14} the energy measured by a stationary observer at radius rr, which is
with (e_(t))^( hat(t))=(1-(2M)/(r))^((1)/(2))\left(\boldsymbol{e}_{t}\right)^{\hat{t}}=\left(1-\frac{2 M}{r}\right)^{\frac{1}{2}}. Recall that p_(t)p_{t} is conserved along the geodesics, so ^(15)p^(t)(r)=ℏomega_(oo)(1-2M//r)^(-1){ }^{15} p^{t}(r)=\hbar \omega_{\infty}(1-2 M / r)^{-1}. This yields
which is simply eqn 22.16 for the gravitational redshift rearranged. The final thing the astronaut sees is a tiny chink of blue light. Bon voyage.
25.5 Gravitational collapse
How are black holes created? When a main-sequence (i.e. a fairly average-sized) star exhausts its nuclear fuel, it generally expands to form a red giant, and subsequently collapses under its own gravitational attraction to form a white dwarf star. However, when a star of mass ≳1.4M_(o.)\gtrsim 1.4 M_{\odot} uses up its nuclear fuel, ^(16){ }^{16} then things can proceed differently.
If the star remains massive enough (i.e. it does not shed its outer layers of mass) it does not achieve the equilibrium state of a stable white dwarf. Instead, it will continue to collapse. If massive enough it contracts through the Schwarzschild radius r=2Mr=2 M as shown in Figs. 25.8 and 25.9. It then continues to collapse further until it is compressed to infinite density. That is, its mass is compressed to a point at the origin, with the result being a singularity in spacetime at r=0r=0.
Can an observer ever hope to observe this singularity? Not at all. We saw in the last section that regular light signals from an in-falling astronaut become less frequent (and also highly redshifted) as the astronaut nears r_(S)r_{\mathrm{S}}. This is also true of photons from the surface of a collapsing star. The image of the shrinking star appears to slow and darken. It freezes, becoming completely black in appearance at its outer surface approaches r_(S)=2Mr_{\mathrm{S}}=2 M from above. The light cones of the photons emitted from the surface of the star are shown in Figs. 25.8 and 25.9. Light emitted as the surface of the star passes through r_(S)r_{\mathrm{S}} has an ingoing wavefront that falls into the singularity at r_(S)=0r_{\mathrm{S}}=0 and an outgoing wavefront that remains at the radius r_(S)=2Mr_{\mathrm{S}}=2 M for ever. Light emitted as the surface collapses further is drawn in towards the black hole, no matter whether it was directed towards larger or smaller rr. (As explained in the next chapter, this curious situation leads to the formation of a so-called closed trapped surface.) At this point, the star itself is something of an irrelevance for the outside world which can no longer interact with it in any way. It is as if the star has now entered a Universe of its own. What matters for the outside Universe is now only the event horizon and the exterior geometry. By the same token, the singularity is hidden from the outside world.
Recall from Chapter 19 that we can map the presence of infinities and singularities in a spacetime using a Penrose diagram to depict the conformal structure. The conformal structure of spacetime that follows from the gravitational collapse that results in a black hole is shown in the Penrose diagram in Fig. 25.10. The black-hole singularity at r=0r=0 is shown by the double line, along with the event horizon H at r_(S)=2Mr_{\mathrm{S}}=2 M. The black-hole singularity is a spacelike surface (just as we had for the t=0t=0 point in the Robertson-Walker Penrose diagram in Chapter 19.) The reason for this is the swapped roles of time and space inside the event horizon. This means we describe r=0r=0 as a spacelike singularity.
For travellers on timelike geodesics that end up at i^(+)i^{+}all of spacetime is visible, except the region for which r < 2Mr<2 M. For those unlucky enough to have fallen through the event horizon, all light rays (which, remember, travel at 45^(@)45^{\circ} in these diagrams) will meet the singularity. There is a future horizon at the singularity: the observer in the region r < 2Mr<2 M cannot access photons from the whole of the space. The Penrose diagram in Fig. 25.11 also shows the surface of a star as it collapses, passing though r_(S)r_{\mathrm{S}} and eventually meeting the singularity.
Fig. 25.9 Gravitational collapse with one spatial dimension suppressed.
Fig. 25.10 Penrose diagram showing the structure of spacetime containing a black hole with event horizon HH.
Fig. 25.11 Penrose diagram showing the gravitational collapse of a star (hatched area) to form a black hole.
Chapter summary
Black holes are regions of spacetime where gravitational curvature doesn't allow light to escape. The spacetime of spherically symmetric black holes is described by the Schwarzschild metric.
The Schwarzschild radius r_(S)=2Mr_{\mathrm{S}}=2 M represents the event horizon for the black hole. Inside the event horizon, the light cone structure means that all matter will inevitably meet the singularity in spacetime at r=0r=0.
An astronaut will not notice anything special when passing the point r=r_(S)r=r_{\mathrm{S}} since this does not represent a physical singularity in spacetime. They will experience huge radially stretching gravitational forces as they approach r=0r=0.
Although it takes an astronaut a finite proper time to meet the singularity, the coordinate time diverges.
Exercises
(25.1) Consider the Schwarzschild metric in the form
where v^(2)=2M//rv^{2}=2 M / r and dOmega^(2)=dtheta^(2)+sin^(2)thetadphi\mathrm{d} \Omega^{2}=\mathrm{d} \theta^{2}+\sin ^{2} \theta \mathrm{~d} \phi.
(a) Compute the metric that results from a coordinate transformation dt=dT-R(r)dr\mathrm{d} t=\mathrm{d} T-R(r) \mathrm{d} r, where R(r)R(r) is a function of rr only.
(b) What function R(r)R(r) is required to give g_(rr)=1g_{r r}=1 ?
(c) Using the function from (b), show that the metric becomes
(25.20)
(d) Comment on the form of the metric describing a hypersurface of constant TT.
(e) What are the velocities dr//dtau\mathrm{d} r / \mathrm{d} \tau and dr//dT\mathrm{d} r / \mathrm{d} T for a radial plunge in these coordinates.
Hint: Make use of the radial plunge results from Chapter 22.
(f) How do light cones behave in these coordinates?
(g) Show that the coordinate speed for a falling observer is always less than the coordinate speed of light.
The coordinates used in eqn 25.20 are known as Painlevé-Gullstand coordinates or global rain coordinates and demonstrate that there is no physical singularity at r=2Mr=2 M (see the next chapter). The line element in these coordinates was independently proposed in 1922 by Paul Painlevé (twice Prime Minister of the French Third Republic) and Allvar Gullstrand (winner of the Nobel Prize in Physiology in 1911). Both were concerned that the solution to the Einstein equation they had discovered showed that relativity allowed infinite numbers of solutions, and so was incomplete. Lemaitre showed in 1933 that these newly discovered solutions were simply the results of a coordinate transformation of the Schwarzschild line element, as the problem demonstrates. Incidentally, Gullstand had also blocked Einstein from receiving the Nobel Prize in Physics for his formulation special relativity in 1905, as he believed that theory to be incorrect.
(25.2) We shall derive, and justify the name of, the global rain coordinates from the previous problem using an array of in-falling clocks. See the books by Taylor, Wheeler, and Bertschinger, and by Moore (whose approach we follow here) for more details.
The synchronized clocks are dropped at a steady rate from rest at infinity (where their readings coincide with the Schwarzschild coordinate time tt ), and fall radially into the black hole. Observers travel with the falling clocks and when they coincide with an event, they read off the time t^(˘)\breve{t} and record the value of r,thetar, \theta and phi\phi. To compute the metric line element in these coordinates we note that t^(˘)=t^(˘)(r,t)\breve{t}=\breve{t}(r, t) and so
Next, we want to work out ((del(t^(ˇ)))/(del r))\left(\frac{\partial \check{t}}{\partial r}\right), which we interpret as the difference in t^(˘)\breve{t} measured by two clocks, separated by a distance dr\mathrm{d} r evaluated a coordinate time tt.
(b) Consider a pair of events separated by dr\mathrm{d} r that occur at the same tt. Show that for the observers attached to the clocks at these events
How can this metric be used to argue that objects following timelike geodesics inside the event horizon must move inwards?
(25.4) A particle in the Schwarzschild geometry around a black hole starts in the theta=pi//2\theta=\pi / 2 plane at radius rr, moving purely tangentially with a local velocity of magnitude v_(0)v_{0}, as measured by a stationary observer.
(a) Using the fact that the relative velocity vv for observers with velocities u\boldsymbol{u} and v\boldsymbol{v} is determined by u*v=-gamma(v)\boldsymbol{u} \cdot \boldsymbol{v}=-\gamma(v), determine the components of the particle's velocity v\boldsymbol{v} in Schwarzschild coordinates at rr.
(b) Assuming the particle moves freely, what are its values of the constants of the motion tilde(E)\tilde{E} and tilde(L)\tilde{L} in terms of v_(0)v_{0} ?
(c) If r=4Mr=4 M, will a value of v_(0)=1//sqrt2v_{0}=1 / \sqrt{2} be sufficient to prevent the particle falling into the hole? See Blennow and Ohlsson for a more complete discussion of this problem.
↷\curvearrowright The material in this chapter provides some background to the Newtonian theory of gravitation. Readers eager to gravitation. Readers eager to eral relativity right away can skip to the next chapter.
